Sharedwww / ribetofficial.dviOpen in CoCalc
����;� TeX output 1996.09.25:2033�������K����b�{�K�����W&��Jc�D��t�qG�cmr17�Scrib���e�B�notes�for�Ken�Rib�et's�Math�274��5?莎��*��K����b�{�K������{��$��N�G� cmbx12�Jan��u�uary�z�17,�1996��b#��X�Q cmr12�Scrib�S�e:�8�La��rwren��Smithline,��%߆�T cmtt12�<[email protected]>��r ��Here��are�some�topics�to�b�S�e�discussed�in�this�course:�� S�����fd�����Galois��represen��rtations�and�mo�S�dular�forms,���������Hec��rk�e��algebras,��������mo�S�dular��curv��res,�and�Jacobians�(Ab�elian�v��X�arieties).������ R�This�\olecture�is�a�brief�o��rv�erview�\oof�some�connection�b�S�et��rw�een�\othese�concepts,���and�also�an����exercise��in�name-dropping.������W��Ve�̓can�describ�S�e�an�an�elliptic�curv��re,�җor�the�Jacobian�of�a�higher�gen�us�curv�e,�җor�ab�S�elian���v��X�ariet��ry��using�a�lattice.�8�F��Vor����g� cmmi12�E���,�the�lattice�is��L�UR�=��H�����|{Ycmr8�1����(�E��(��,��N� cmbx12�C�� ��)�;����Z�)��,���!",� cmsy10�!��C�;�꨹b��ry�the�map�� ��n�7!��UR��q����u cmex10�R���6�=��2cmmi8� ��g޼!�n9:������W��Veil�&�considered�curv��res�o�v�er�a� nite� eld,�u��k����of�c�haracteristic��p�.���There�is�an�algebraic���de nition��of��L=nL��for��n�UR���1�;��ꦹgcd����(�n;���p�)�=�1�:��縍�r��E���[�n�]�UR=��f�P���2��E��(����W��Q���*���k������)�:��nP���=�0�g��=�����Fu�����1�����콉fe(P��'��n����� ��L=L��=��L=nL:����F��Vor��example,�let��n�UR�=������2��� :�for�꨼����1,�and����a�prime�di eren��rt�from��p:�������W��Veil��further�considered�the�limit��ق�����E���[���� ���K� cmsy8�1�� �]�UR=��������U�1�� ���� �����[��� ҁ������=1����^�E��[���� ����3��]�:��#/{��T��Vate��ob�S�erserv��red�there�is�a�map��E���[�����2���3��]������������g ��UR�!�������E��[�����2������1���]�;�꨹and�so�of�course�this�in�v�erse�limit,��������lim�������� ����g��E���[���� ����3��]�UR=��T���̽����(�E��)��� �is��called�the�T��Vate�mo�S�dule.������As��#�E���[�n�]�is�free�of�rank�2�o��rv�er��#�Z�=n�Z�,��so�is��T���̽����(�E��)�free�of�rank�2�o��rv�er��#�Z���̽����.�_PAlso,���V���̽l��!ȹ(�e�)���=����T���̽����(�E���)�V�� ��Q���̽�����is���a�2�dimensional�v��rector�space�o�v�er��Q���̽����.�+8This�is�the� rst�example�of���-adic�n!��s�etale���cohomology��V.�8�F�or��top�S�ological�space��X��,��X�Fտ7!�UR�H����2���V�i��RA����Q�����e����pt����:�(�X���=����W��Qȹ��*���k�����;����Q���̽����)�:������Here��are�the�names�of�some�co�S�ol�folks:�8�T��Vaniy��rama�Shim�ura�Mumford�T��Vate.������No��rw,�� elliptic�ޢcurv�e��E��=�Q��gets�an�action�of��G�UR�=��G��.�al�C��(����!�����Q��� #��=�Q�),�� and�so�do�S�es��E���[�n�].�4�I.e.���n9�(�P�3߹+����Q�)���=���n9�(�P��ƹ)��+����(�Q�)�:�D�So�w��re�ha�v�e�a�homomorphism�����:��G��!���Aut��֩(�E���[�n�])�=��GL���̺2����(�Z�=n�Z�)�:�D�Since���w��re��ha�v�e�an�exact�sequence������f1�UR�!���k��rer������!��G��!���im��c����!��0�;��u���w��re��get�a�to�w�er�of� elds��������$��������Q����{��!�UR�K�1�!��Q�;����and�꨿G��.�al�C��(�K�5�=�Q�)�UR=��im��c������GL���̺2����(�Z�=n�Z�)�:��������And���no��rw�for�something�completely�dieren�t.��W��Ve�can�also�get�to�these�Galois�represen�ta-���tions�!Fvia�mo�S�dular�forms.�ܹLet��k��c�b�e�the�w��reigh�t,�.�suc�h�!Fas�2.�ܹLet��N�b*�b�e�the�lev��rel.�ܹThe�complex���v��rector�Cspace��S���̽k��#��(�N�@�)�is�the�set�of�cusp�forms�on����̺1����(�N��),�gja�nite�dimensional�v��rector�space,���namely��V,��the�set�of�holomorphic�functions��f�2��on��/�%n�
eufm10�H��suc��rh�that��縍���f�G��((�az�3��+����b�)�=�(�cz��+��d�))�UR=�(�cz�3��+��d�)�����k��#��f�G��(�z���)�������1�������K����b�{�K������{��for��	������͟�f\� �����d�����y�a����L�b��������!�c�����zd�������|��f\�!����z�2�UR����̺1����(�N�@�)�;�	�Na;���d����1�;�c����0(�N�@�)�:���Suc��rh��an��f�2��has�a�p�S�o�w�er�series�(or�F��Vourier�series)�expansion�in��q�Ë�=��URexp��H�(2��n9iz���):���ݍ��(�f�G��(�z���)�UR=�������N3�1��
��������X���ҁ��n5�1�����c���̽n���P�q��n9����n����:�����Here���is�a�famous�example�observ��red�b�y�Raman�ujan,��cand�pro�v�ed�b�y�Mordell�using�(his)�Hec�k�e����op�S�erators:�������׼q����������1��
���n7�����Y���ҁ���ĺ1���5U�(1������q��n9����n����)�����24��UZ�=�������N3�1��
���UR�����X���ҁ��n5�1������W�(�n�)�q��n9����n���;���e��for���Raman��rujan's�����function.��No�w,��x��W�(�n�)���(�m�)�i}=���W�(�nm�)���for��gcd��,�(�n;���m�)�i}=�1.��Also,�there���is�a���recurrence��for�prime�p�S�o��rw�ers.�8�Am�usingly��V,��the�normalized�basis�elemen��rt�of��S���̺12��	�(1)�is�����c��UR=�������N3�1��
��������X���ҁ��n5�1������W�(�n�)����exp���(2��n9inz���)�:��q荹Ev��ren��more�am�usingly��V,������ռ�W�(�n�)�UR����������X���������d�j�n�����d�����11���X�(691)�:������L����Exp�S�erience��and�Shim��rura�ha�v�e�sho�wn�that�there�exist��f��Q�2�UR�S���̽k��#��(�N�@�)�suc�h�that������^��T���̽n���P�(�f�G��)�UR=��c���̽n��R������f����for��all��n�UR���1��for�some�scalars��c���̽n���P�,�and�����5̼f��Q�=����UR�����X������c���̽n���P�q��n9����n�����for���the�same��c���̽n���P�,���and�that�these��c���̽n��	9�are�algebraic�in��rtegers�in�a�nitely�generated�n�um�b�S�er�eld.����That��is,�[�Q�(�c���̽n�����:�UR�n����1)�:��Q�]��is�nite.������Ho��rw��can�w�e�study�and�in�terpret�this?�8�W��Ve�start�with�the�Hec�k�e�ring,�������-�Q�[�T���̽n���P�]�UR���E��nd�(�S���̽k��#��(�N�@�))�:����Serre�kin�1968�said�there�should�b�S�e�Galois�represen��rtations�attac�hed�to�forms�of�arbitrary����w��reigh�t.�	OADeligne�Gsconstructed�them.�In�a�broad�strok��re,���one�can�sa�y�that�w�e�get�b�S�et�w�een���Galois��represen��rtations�and�mo�S�dular�forms�via�F��Vrob�enius�elemen��rts.������Next��time,�w��re�con�tin�ue�with�the�semihistorical�o�v�erview.��'���Jan��u�uary�z�19,�1996��b#��Scrib�S�e:�8�William��Stein,��<[email protected]>��a/��2��N�ffcmbx12�Mo�s3dular�ffRepresen���tations�and�Mo�dular�Curv���es�������2�������K����b�{�K������{��Arithmetic�ffof�Mo�s3dular�F���forms��@��Supp�S�ose����f��L�=���M�����P����*���>��1���̍�>��n�=1���!�R�a���̽n���P�q��n9���2�n��
��is�a�cusp�form�in��S���̽k��#��(�N�@�)�whic��rh�is�an�eigenform�for�the�Hec�k�e�op-����erators.�z�The��Mellin�transform�asso�S�ciates�to��f�H��the��L�-function��L�(�f���;���z���)�.�=�������P����*������1���̍��Ľn�=1���"A��n����2��s���
��a���̽n�����.�Let����K�1�=�UR�Q�(�a���̺1����;���a���̺2���;��:�:�:��ʞ�).�One�j_can�sho��rw�that�the��a���̽n��	��are�algebraic�in�tegers�and��K�F��is�a�n�um�b�S�er�eld.���When�Z��k�z͹=��2,�vu�f���is�asso�S�ciated�to��f��an�ab�S�elian�v��X�ariet��ry��A���̽f��	џ�o�v�er��Q��of�dimension�[�K��N�:���Q�],�vuand����A���̽f��
(~�has��_a��K����action.��(See�Shim��rura,��
�3���@cmti12�Intr��ffo�duction���to�the�A��2rithmetic�The��ffory�of�A�utomorphic���F���unctions�,��Theorem�7.14.)�������Example��1�(Mo�dular�Elliptic�Curv��es)����2�If�)��a���̽n�����2�UR�Q��for�al���l��n�,�+�then��K�1�=��Q��and��[�K�1�:��Q�]�=���1�.��hIn�d5this�c��ffase,�pt�A���̽f��	�T�is�a�one�dimensional�ab�elian�variety,�ptwhich�is�an�el���liptic�curve,�sinc��ffe�it���has�35nonzer��ffo�genus.�fiA��2n�el���liptic�curve�arising�in�this�way�is�c�al���le�d�mo�dular.������Denition��1���K���El���liptic��[curves��E���̺1��	p_�and��E���̺2���ar��ffe��isogenous��if�ther�e�is�a�morphism��E���̺1����!�=�E���̺2��	p_�of���algebr��ffaic�35gr�oups,�which�has�a�nite�kernel.������The��follo��rwing�conjecture�motiv��X�ates�m�uc�h�of�the�theory��V.������Conjecture��1���Q�Q�Every�_el���liptic�curve�over��Q��is�mo��ffdular,�J)that�is,�iso��ffgenous�to�a�curve�c�on-���structe��ffd�35in�the�ab�ove�way.������F��Vor�2��k�6���Ĺ2,�D�Serre�and�Deligne�found�a�w��ra�y�2�to�asso�S�ciate�to��f�z��a�family�of���-adic�represen-���tations.�8�Let�꨼��b�S�e�a�prime�n��rum�b�er��and��K��F�b�e�as�ab�o��rv�e.�8�It��is�w��rell�kno�wn�that������av�K��F�
�����-2�@�cmbx8�Q��
��Q���̽������P���N8����԰���g �=��������������Y����������j����"���K���̽��uZ�:��%��One��can�asso�S�ciate�to��f�2��a�family�of�represen��rtations�����:>����̽;f���˹:�UR�G��=��G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)��!���GL����(�K��F�
�����Q��
��Q���̽����)���unramied���at�all�primes��p����6�j�N�@�.���By���unramied�w��re�mean�that�for�all�primes��P�O��lying�o�v�er�����p�,�mthe�R�inertia�group�of�the�decomp�S�osition�group�at��P��is�con��rtained�in�the�k�ernel�of���.�q�(The���decomp�S�osition��rgroup��D���̽P��
�>�at��P�i8�is�the�set�of�those��g�Ë�2�UR�G��whic��rh�x��P��and�the�inertia�group�is���the��k��rernel�of�the�map��D���̽P��
g�!�URG��.�al�C��(�O�UV�=P��ƹ),�where��O�?��is�the�ring�of�all�algebraic�in�tegers.)������No��rw�Z�I���̽P��
����{�D���̽P����G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)�Zand��D���̽P��̼=I���̽P��&�is�cyclic,�since�it�is�isomorphic�to�a�subgroup�of���the���galois�group�of�a�nite�extension�of�nite�elds.�+�So��D���̽P��̼=I���̽P��
Ԕ�is�generated�b��ry�a�F��Vrob�S�enious���automorphism���frob���z���̽p��!,��lying��o��rv�er��p�.�8�W��Ve�ha�v�e���������tr���j(����̽;f��Ly�(�frob������̽p��WO�))�UR=��a���̽p����2��K�1���K��F�
����Q���̽�����and��������*�det���|�(����̽����)�UR=��������k�6���1���%������";������;�(0.1)�����where�꨼���̽��㎹is�the���th�cyclotomic�c��rharacter�and��"��is�a�Diric�hlet�c�haracter.�������3�������K����b�{�K������{��Characters���	��Let�꨼f��Q�2�UR�S���̽k��#��(�N�@�).�8�F��Vor�all�����f\� �����d����T�a���!�ab�������c�c���!Ud�����,W��f\�!���7[U�2���SL���3ݟ�̺2����(�Z�)�with��c�����P�������԰���n:�=�������0��mo�S�d��o�N�+��w��re�ha�v�e��+���r��f���G���f\� �������ō�eܼaz�3��+����b��eܟQm�fe�����-cz�3��+����d������-:��f\�!���8z�=�UR(�cz�3��+����d�)�����k��#��"�(�d�)�f�G��(�z���)�;��"
��where��j�"�UR�:�(�Z�=n�Z�)����2���V�!��C����2����n�is�a�Diric��rhlet�c�haracter�mo�S�d��N�@�.��vIf��f�5i�is�an�eigenform�for�the�diamond����brac��rk�et��op�S�erator��<�URd�>�,��(so�that��f�G��j�UR�<�d�>�=��"�(�d�)�f��)��then��"��actually�tak��res�v��X�alues�in��K�ܞ�.������Let��d����̽n��
X��b�S�e�the�mo�d��n��cyclotomic�c��rharacter.��The�map�����̽n��
N4�:���G��!��(�Z�=n�Z�)����2���	ph�tak�es��d�g��2��G����to�5�the�automorphism�induced�b��ry��g���on�the��n�th�cyclotomic�extension��Q�(����̽n���P�)�of��Q�,�Huwhere�w�e���iden��rtify�꨿G��.�al�C��(�Q�(����̽n���P�)�=�Q�)�with�(�Z�=n�Z�)����2�����.�8�The��"��app�S�earing�in�(0.1)�is�really�the�comp�osition�������G�����K��� ����;�cmmi6�n���������UR�����
��!�������(�Z�=n�Z�)��������������ɜ�"��g	�����V�����j�!������C���������:�������F��Vor��eac��rh�p�S�ositiv�e�in�teger����o�w�e�consider�the������2���3��th�cyclotomic�c�haracter�on��G�,�����U����̽���������Ĺ:�UR�G��!��(�Z�=�������3��Z�)���������:����Putting��these�together�giv��re�a�map����������̽������q�%cmsy6�1���#��=���URlim��
�{����G���� �������������⍍�	$g���������� � ���̽�������� �Ĺ:�UR�G�����K���7� ��i?��������������� ��!�������Z��� ������ڍ������:��'�%��P���arit�y�ffConditions��@��Let� X�c�]��2�G��.�al�C��(�����fe #�� n��Q��� #��=�Q�)�b�S�e�complex�conjugation.���W��Ve�ha��rv�e�� ���̽n���P�(�c�)�]�=���1,�h�so��"�(�c�)�=��"�(��1)�and����� ���̽����(�c�)�UR=�(��1)����2�k�6���1���.�8�Let�� �����* ��f\� �����d����ȼa������b���������c����?�d������Z˟�f\�!���ښɹ=���UR��f\� �����d���?���1���. �0��������0���)uQ��1�����=����f\�!���G�N�:�� � ��F��Vor�꨼f��Q�2�UR�S���̽k��#��(�N�@�),�����x�f�G��(�z���)�UR=�(��1)��� ��k��#��"�(��1)�f��(�z���)�;����so��(��1)����2�k��#��"�(��1)�UR=�1.�8�Th��rus,�������Fdet�����(����̽����(�c�))�UR=���(��1)(��1)��� ��k�6���1��U�=���1�:����The���det��' c��rharacter��is�o�S�dd�so�the�represen�tation�����̽��㎹is�o�S�dd.�������Remark��1���?�6�(V���ague�j Question)�How�c��ffan�one�r�e�c�o�gnize�r�epr�esentations�like�����̽;f�����\in�natur�e"?����Mazur��Hand�F���ontaine�have�made�r��ffelevant�c�onje�ctur�es.�9�The�Shimur�a-T���aniyama�c�onje�ctur�e�c�an���b��ffe��r�eformulate�d�by�saying�that�for�any�r�epr�esentation�����̽;E�����c�omming�fr�om�an�el���liptic�curve��E����ther��ffe�35is��f�{4�so�that�����̽;E������P���X4����԰���q �=����� ݼ���̽;f�� Ly�.��� ����4����'��K����b�{�K������{��Conjectures�ffof�Serre�(mo�s3d����g�ff cmmi12���v���ersion)��@��Supp�S�ose����f��is�a�mo�dular�form,��E���a�rational�prime,����a�prime�lying�o��rv�er�����,�and�the�represen��rtation���s��������̽;f�� ?�:�UR�G��!���GL������̺2����(�K���̽��uZ�)���(constructed�Z�b��ry�Serre-Deligne)�is�irreducible.� �gThen�����̽;f��#m�is�conjugate�to�a�represen�tation����with���image�in��GL���f���̺2��&��(�O���̽��uZ�),��where��O���̽�� LD�is�the�ring�of�in��rtegers�of��K���̽���.���Reducing�mo�S�d����giv��res�a���represen��rtation������S9���&�fet�]ڍ������b���T�;f��΀�:�UR�G��!���GL������̺2����(�F���̽��uZ�)���V�whic��rh��8has�a�w�ell-de ned�trace�and�det,�Ȝi.e.,�the��8det�and�trace�don't�dep�S�end�on�the�c�hoice���of���conjugate�used�to�reduce�mo�S�d���.�~�One�kno��rws�from�represen�tation�theory�that�if�suc�h�a���represen��rtation�3is�semisimple�then�it�is�completely�determined�b�y�its�trace�and�det.�Th�us�if�������&�fet�]ڍ�����t��T�;f��v��is���irreducible�it�is�unique�in�the�sense�that�it�do�S�esn't�dep�end�on�the�c��rhoice�of�conjugate.������W��Ve��ha��rv�e�the�follo�wing�conjecture�of�Serre�whic�h�remains�op�S�en.��>�����Conjecture��2�(Serre)����5�A��2l���l��Oirr��ffe�ducible�r�epr�esentation�of��G��over�a� nite� eld�which�ar�e�o�dd,���i.e.,�R �det�(��n9�(�c�))��@=���1�,��c���c��ffomplex�c�onjugation,�R ar�e�of�the�form�����&�fet�]ڍ����� (!��T�;f�� ��for�some�r�epr�esentation�������̽;f���"�c��ffonstructe�d�35as�ab��ffove.��~;����Example��2���DUQ�L��ffet����E��=�Q��b�e�an�el���liptic�curve�and�let�����̽��2��:�9��G��!���GL����V��̺2���Z�(�F���̽����)����b�e�the�r�epr�esentation���induc��ffe�d�Waby�the�action�of��G��on�the���-torsion�of��E���.���Then���det��� ����̽���4�=��N� ���̽��PG�is�o��ffdd�and�����̽���is�usual���ly���irr��ffe�ducible,��so�WSerr�e's�c�onje�ctur�e�would�imply�that�����̽��Pe�is�mo�dular.��GF���r�om�this�one�c�an,��under���Serr��ffe's�35c�onje�ctur�e,�pr�ove�that��E��L�is�mo�dular.������De nition��2���K���L��ffet��n��Ë�:�UR�G��!���GL������̺2����(�F�)��(�F��is�a� nite� eld)�b��ffe�a�r�epr�esenation�of�the�galois�gr�oup����G�.�_�The� �we�say�that�the��represen��rtions�Մ��C��is�mo�S�dular� ��if�ther��ffe�is�a�mo�dular�form��f�G��,�#�a�prime���,���and�35an�emb��ffe�dding�35�F�UR�,���!������fe|r� n��F���� �ğ�̽��zS�such�that�������P���Ë����԰����s�=������n4���&�fet�]ڍ�����}���T�;f��'y��over������fe|r� n��F���� ����̽��%�.��"g>��Wile's�ffP���ersp�s3ectiv�e��@��Supp�S�ose��0�E��=�Q��is�an�elliptic�curv��re�and�����̽;E��X4�:�UR�G��!���GL������̺2����(�Z���̽����)��0the�asso�ciated���-adic�represen��rtation���on��the�T��Vate�mo�S�dule��T���̽����.�8�Then�b��ry�reducing�w�e�obtain�a�mo�S�d����represen�tation���s�����U���&�fet�]ڍ�������ɟ�T�;E������=�UR����̽;E��X4�:��G��!���GL������̺2����(�F���̽����)�:����If��w��re�can�sho�w�this�is�mo�S�dular�for�in nitely�man�y����then�w�e�will�kno�w�that��E����is�mo�S�dular.������Theorem��1�(Langlands�and�T���unnel)���פ��If�]o����̺2�;E��'o�and�����̺3�;E���ar��ffe�irr�e�ducible,��0then�they�ar�e�mo�d-����ular.������This�kis�pro��rv�ed�kb�y�using�the�fact�that��GL�������̺2�����(�F���̺2����)�and��GL�������̺2���(�F���̺3����)�are�solv��X�able�so�w��re�ma�y�apply���\base-c��rhange".������Theorem��2�(Wiles)���uƼ�If����is�an���-adic�r��ffepr�esentation��which�is�irr��ffe�ducible��and�mo��ffdular�mo�d�����35�with���UR>��2�35�and�c��ffertain�other�r�e�asonable�hyp�othesis�ar�e�satis e�d,�then����itself�is�mo�dular.��� ����5����4<��K����b�{�K������{��Jan��u�uary�z�22,�1996��b#��Scrib�S�e:�6iLa��rwren��Smithline,���<[email protected]>��(Note.�7;These�w�ere�done�ab�S�out�t�w�o�mon�ths�after����the��fact,�since�the�assigned�p�S�erson�didn't.�8�So�they�are�terse.�It�w��ras�a�prett�y�easy�lecture.)��둍���T��Vo�S�da��ry�,�w�e��$ha�v�e�a�limited�goal:���to�explain�mo�S�dular�forms�as�functions�on�lattices�{�or���elliptic��0curv��res.�,�(See�Serre's��Course��in�A��2rithmetic�,��{or��0Katz's�pap�S�er�in�the�Pro�ceedings�of�the���An��rt�w�erp��Conference�in�the�Springer�LNM�series.)������Let��the�lev��rel��N��6�=�UR1.�8�Consider�a�w�eigh�t��k�QŹcusp�form��f�G��.�8�F��Vor����o�2�UR�H�,�w�e�ha�v�e���>����q�f�G��(�a�Ź+����b=c��+��d�)�UR=�(�c�Ź+��d�)�����k��#��f�G��(��W�)�:����So�;�f�G��(���p�+�hS1)�/}=��f��(��W�).�|�By�;the�map������7!�/}�q����=��exp��"�(2��n9i��),�F�w��re�;map��H��to�the�punctured�disc�����f�z���:�t
0��<��j�z���j��<��1�g�.�o
W��Ve���abuse�notation�and�think�of��f�D��as�a�function�on�this�disc.�Since��f�D��is�a���cusp��form,��f�2��extends�to�0,�and��f�G��(0)�UR=�0.�8�So��w��re�ha�v�e�a��q�n9�-expansion������<L�f��Q�=���������1��
���ԟ����X���ҁ��UR�n�=1���Z�a���̽n���P�q��n9����n����:��(4���Lattices�ffinside�C��@��Let���L�b��=��Z�!���̺1���i��e�Z�ܞ!���̺2����.��W��Ve�ma��ry�assume�that��!���̺1���=!���̺2��	"��2�b��H�.��Let��R��b�S�e�the�set�of�lattices�in��C�.����SL���
ދ��̺2�����Z��5�acts�on�the�left�of��M��6�=�UR�f�(�!���̺1����;���!���̺2���)�:��!���̺1���;���!���̺2��V�2��H�g��5�b��ry�m�ultiplication�of�the�column�v�ector.���This��action�xes�the�lattice.������Here��is�the�relation�with�elliptic�curv��res.�8A���lattice��L��determines�a�complex�torus��C�=L�.���There��xis�a�W��Veierstrass��}��function�on�this�torus.�1�Consider�an�elliptic�curv��re��E����o�v�er��C�.�1�There���is��a�lattice�giv��ren�b�y�the�inclusion��H���̺1����(�E���(�C�)�;����Z�)���,���!��C�.��5Cho�S�ose��a�nonzero��!�CG�2���H���V���2�0���Z�(�E�;����
����2��0��RA��E���-��)�:����Then�꨼
��n�2�UR�H���̺1�����maps�to����qīR������=�
���4�!�Ë�2��C�.������So��ma��ryb�S�e�w�e�should�think�of��R��as�the�set�of�pairs��f�(�E��;���!�n9�)�g�:������W��Ve��cha��rv�e�a�map��M���=�C����2���
qʿ!�UR�H��sending�(�!���̺1����;���!���̺2���)�to��!���̺1���=!���̺2���:��No��rw�tak�e�the�quotien�t�on�the�left���b��ry���SL����3��̺2���7�Z�:�����C�R�=�C�����times��0f�=��URSL���3ݟ�̺2�����Z�n�M���=�C����������
qʿ�����#!��UR�SL���3ݟ�̺2���Z�n�H�:��Э��But�Bthis�just�is�the�space�of�elliptic�curv��res�o�v�er��C�.�	@	So��f��^�:��_�H��!��C�B�whic�h�is�a�mo�S�dular���form�Laand��F��n�:����M�<��!��C�La�satisfying��F��ƹ(�L�)���=������2��k���
�F��(�L�)�Laamoun��rt�to�the�same�thing�b�y�a�simple���calculation.��"b1��Hec���k�e�ffOp�s3erators����Let�꨼F��n�b�S�e�a�function�on�lattices.�8�Dene�the�Hec��rk�e��op�erator��T���̽n��	���as���>�����T���̽n���P�F��ƹ(�L�)�UR=����95�����X��������L������0������L;�(�L�:�L������0���)=�n���?�޼F��(�L�)�n�����k�6���1���:��"������The��Cessen��rtial�case�on�elliptic�curv�es�is�for��n�UR�=����C�a�prime.�8In�this�case,��the��L����2�0���|�corresp�S�ond���to��the������+�1��subgroups�of�order����of�(�Z�=�Z�)����2�2����.�������6����CY��K����b�{�K������{��Jan��u�uary�z�24,�1996��b#��Scrib�S�e:�8�William��Stein,��<[email protected]>��+��More�ffOn�Hec���k�e�ffOp�s3erators���č���W��Ve�s$consider�mo�S�dular�forms��f��#�on����̺1����(1)��=��SL������̺2����(�Z�),��Cthat�s$is,�holomorphic�functions�on�����H����[�f1g�꨹whic��rh�satisfy��s����ѯ�f�G��(��W�)�UR=��f��(������ō�33�a�Ź+����b��33�Qm�fe ,���-c�Ź+����d�����"z��)(�c�Ź+����d�)������k��� &<��for��all�����f\� �����d����T�a���!�ab�������c�c���!Ud�����,W��f\�!���7[U�2��UR�SL���3ݟ�̺2����(�Z�).�8�Using�a�F��Vourier�expansion�w��re�write��$�3��� ��f�G��(� �W �)�UR=���������1�� ��� ԟ����X��� ҁ���n�=0���Z �a���̽n���P�e��� ��2��I{i ��rn���^�;�� S4��and��_sa��ry��f��^�is�a�cusp�form�if��a���̺0��V�=�UR0.�"rThere�is�a�corresp�S�ondence�b�et��rw�een��_mo�dular�forms��f��^�and���lattice��functions��F��n�satisfying��F��ƹ(�L�)�UR=������2��k�� � �F��(�L�)��giv��ren�b�y��F��ƹ(�Z� �Ź+����Z�)�UR=��f�G��(� �W �).��!�U��Explicit�ffDescription�of�Sublattices��@��The�꨼n�th�Hec��rk�e��op�S�erator��T���̽n�� ���of�w��reigh�t�꨼k�QŹis�de ned�b��ry���_����˼T���̽n���P�(�L�)�UR=��n��� ��k�6���1�����)�C�����X��� ���� �L������0������L;�꨺(�L�:�L������0���)=�n���[email protected]�L��� ��0���9�:�� � ��What��are�the��L����2�0����explicitly?�8�Note�that��L=L����2�0���is�a�group�of�order��n��and�����l��L��� ��0���9�=nL�UR���L=nL��=�(�Z�=n�Z�)��� ��2����:����W��Vrite�ƥ�L���=��Z�!���̺1�����+���Z�!���̺2����,�=�let��Y���̺2�� ���b�S�e�the�cyclic�subgroup�of��L=L����2�0���޹generated�b��ry��!���̺2���and�let�����d��ȹ=�#�Y���̺2����.���Let� ��Y���̺1��m̹=�(�L=L����2�0���9�)�=�� Y���̺2���.��ɼY���̺1��ޤ�is� �generated�b��ry�the�image�of��!���̺1���so�it�is�a�cyclic�group�of���order��.�a�UR�=��n=d�.� W��Ve�w��ran�t�to�exhibit�a�basis�of��L����2�0���9�.� Let��!����2��n9�0��RA��2���V�=�UR�d!���̺2���2��L����2�0��ng�and��.use�the�fact�that��Y���̺1�����is�^�generated�b��ry��!���̺1�� �to�write��a!���̺1��V�=�UR�!����2��n9�0��RA��1���M0�+��,�b!���̺2���for��b�UR�2��Z�^�and��!����2��n9�0��RA��1���V�2�UR�L����2�0���9�.� ISince��b��is�only�w��rell-de ned���mo�S�dulo�꨼d��w��re�ma�y�assume�0�UR���b����d������1.�8�Th�us���������z��f\� �����d�����&�!����2��n9�0��RA��1�����������&�!����2��n9�0��RA��2��������5��f\�!����A3�=���UR��f\� �����d���?��a��� � b�������e��0��� j��d�����+���f\�!�����5p���f\ �����d���B[W�!���̺1��������B[W�!���̺2������Slf��f\�!��������and��the�c��rhange�of�basis�matrix�has�determinen�t��ad�UR�=��n�.�8�Since���_���x��Z�!��� ���n9�0���ڍ�1���j��+����Z�!��� ���n9�0���ڍ�2���V��UR�L��� ��0��#����L��=��Z�!���̺1���+����Z�!���̺2�����and��c(�L�UR�:��Z�!����2��n9�0��RA��1����L�+�H�Z�!����2��n9�0��RA��2�����)�=��n��(since�the�c��rhange�of�basis�matrix�has�determinen�t��n�)�and�(�L�UR�:��L����2�0���9�)�=��n�����w��re��see�that��L����2�0��#��=�UR�Z�!����2��n9�0��RA��1���j��+����Z�!����2��n9�0��RA��2�����.������Th��rus���there�is�a�one-to-one�corresp�S�ondence�b�et��rw�een���sublattices��L����2�0��� ����L��of�index��n��and���S�matrices����e(�������d��� �)�a��� Y6b������� Ŭ�0��� �*�d�������(�,�)����1�Uwith�e�ad��&�=��n��and�0����b����d������1.��In�particular,�Twhen��n��&�=��p��is�prime�there����p�T3�+�1��of�these.�#�In�general,�!�the�n��rum�b�S�er��of�suc��rh�sublattices�equals�the�sum�of�the�p�ositiv��re����divisors��of��n�.��� ����7����Q��K����b�{�K������{��Action�ffof�Hec���k�e�ffOp�s3erators�on�Mo�dular�F���forms��@��No��rw���assume��f�G��(� �W �)�j�=�������P����*���� �1���̍�� �m�=0���#�\�c���̽m��ļq��n9���2�m�� ��is���a�mo�S�dular�form�with�corresp�onding�lattice�function��F��ƹ.����Ho��rw��can�w�e�describ�S�e�the�action�of�the�Hec�k�e�op�S�erator��T���̽n�� ���on��f�G��(� �W �)?�8�W��Ve�ha�v�e����(����� ���b\x�T���̽n���P�F��ƹ(�Z� �Ź+����Z�)������=�UR�n����2�k�6���1������������X��� �^�����(����ɛ�����8��O�a;b;d��qȍ�fBab��Aa�cmr6�=�n����6e�� �0��b<d�������*\��F��ƹ((�a �Ź+����b�)�Z��+��d�Z�)���1C퍍����=�UR�n����2�k�6���1����� �����X����"qӼd��� ���k�� � �F��ƹ(������ō�33�a �Ź+����b��33�Qm�fe ,�  �� ��d�����"z��Z����+��Z�)����͍�����=�UR�n����2�k�6���1����� �����X����"qӼd��� ���k�� � �f�G��(������ō�33�a �Ź+����b��33�Qm�fe ,�  �� ��d�����"z��)����<������=�UR�n����2�k�6���1�����Wa�����X��� 8獑 �a;d;b;m���/ }�d��� ���k�� � �c���̽m��ļe��� ��2��I{i�(��������33�a ��ǻ+�b��33�� �fe" �������d��������)�m����%U\�������=�UR�n����2�k�6���1�����W������X��� 8獑 �a;d;m���)!-�d��� ��1��k���c���̽m��ļe�����O��33�2��7 iam ��33�x|�fe������ �d���������ō� ��1�� ��Qm�fe�  ��d�����������' ��d��1�� ���'������X��� 8獑'}�b�=0���6Zֹ(�e�����O��33�2��7 im��33�x|�fezv�������d����ܹ)��� ��b���� 㣍������=�UR�n����2�k�6���1�����P������X���qʍ����c���x��ad�=�n���q�� �m������0������0������)�d��� ��1��k���c���̽dm������0���3�e��� ��2��I{iam���-:�0���ǽ ����#瞍������=���� ������X��� G ��UR�ad�=�n;��m������0������0���5W��a��� ��k�6���1���c���̽dm������0���3�q��n9�� ��am���-:�0������:����������In��cthe�second�to�the�last�expression�w��re�let��m�}�=��dm����2�0���9�,��Ҽm����2�0��KA���0,�then��cused�the�fact�that�the���sum�����Fu��-��1�� ۟콉fe_���'��d������ �������P�������>�d��1���%��>�b�=0���&��(�e����L䍑33�2��7 im��33�x|�fezv�������d����ܹ)����2�b�� ��is��only�nonzero�if��d�j�m�.������Th��rus�����9M�T���̽n���P�f�G��(�q�n9�)�UR=����Ļ�����X��� 8獓�ad�=�n;��m��0���2��a��� ��k�6���1���c���̽dm�� dl�q���� ��am��� c���and��if���UR���0��then�the�co�S�ecien��rt�of��q��n9���2��� ��is���0������fݟ����X��� �����yB�a�j�n;��a�j������?�a��� ��k�6���1���c����s3��33�n��33��u�fe 7��%��sya���G �2����� ��:��%i^����Remark��2���?�6�When�*J�k��o��UR�1��the�c��ffo�ecients�*Jof��q��n9���2��� 5��for�al���l����b��ffelong�to�the��Z�-mo�dule�gener�ate�d�by���the�35�c���̽m����.���\����Remark��3���?�6�Setting�35��UR�=�0��gives�the�c��ffonstant�c�o�ecient�of��T���̽n���P�f�{4�which�is���������d�����X��� �������a�j�n����B+�a��� ��k�6���1���c���̺0��V�=�UR����̽k�6���1���(�n�)�c���̺0����:��%&ލ�Thus�3 if��f�{ �is�a�cusp�form�so�is��T���̽n���P�f�G��.�fb(�T���̽n���f��is�3 holomorphic�sinc��ffe�its�original�de nition�is�as�a��� nite�35sum�of�holomorphic�functions.)���\����Remark��4���?�6�Setting�$��UR�=�1��shows�that�the�c��ffo�ecient�of��q�p]�in��T���̽n���P�f�J#�is�just��c���̽n���.�VAs�an�imme��ffdiate���c��ffor�ol���lary�35we�have�the�fol�lowing�imp��ffortant�r�esult.�������8����	_P��K����b�{�K������{����Corollary��1���H��Supp��ffose�匼f�-��is�a�cusp�form�for�which��T���̽n���P�f��has�0�as�c��ffo�ecient��of��q�S��for�al���l��n�UR���1�,����then�35�f��Q�=�UR0�.��2Q����Remark��5���?�6�When�V�n����=��p��is�prime�we�get�an�inter��ffesting�formula�for�the�action�of��T���̽p��	]�on�the����q�n9�-exp��ffansion�35of��f�G��.�fiOne�has�����r&�T���̽p���]�f��Q�=����2�����X������UR���0������������X�������Nٽa�j�n;�35a�j����5�c�a�����k�6���1���c����s3��33�n��33��u�fe	7��%��sya���G�2�������q��n9������F�:��B��Sinc��ffe�K�n���=��p��is�prime�either��a��=�1��or��a��=��p�.���When��a��=�1�,�R�c���̽p��0Y�o��ffc�curs�in�the�c�o�ecient�of��q��n9���2������and�35when��a�UR�=��p�,�35we�c��ffan�write���UR�=��p�35�and�we�get�terms��p����2�k�6���1���c���̽��	���in��q��n9���2�p��
*��.�fiThus������g��T���̽n���P�f��Q�=����2�����X������UR���0���Nټc���̽p��	�j�q��n9��������+����p�����k�6���1�����������X���8獑���0���#��c���̽��uZ�q��n9����p��
*�:��-v��Jan��u�uary�z�26,�1996��b#��Scrib�S�e:�8�Amo�d��Agashe,��<[email protected]>���(����F��Vollo��rwing�vXthe�notation�of�the�last�few�lectures,���let��M���̽k����denote�the�space�of�mo�S�dular�forms���of��w��reigh�t��k�QŹfor��S���L���̺2����(�Z�)�and��S���̽k��	:�denote�the�subspace�of�cusp�forms.������Then��w��re�ha�v�e:������Prop�osition��1���U���.�#��M���̽k��
@��is��a�nite�dimensional��C�-ve��ffctor�sp�ac�e�and�is�gener�ate�d�by�mo�dular���forms�35having�the�c��ffo�ecients�35of�their�F���ourier�exp��ffansion�in��Q�.����9�-�
cmcsc10�Sketch��lof�Pr���oof�.�B_(F��Vor�C(details,�YHrefer�Serre's�\A�Ccourse�in�Arithmetic"�or�Lang's�\In��rtro-���duction��to�mo�S�dular�forms".)��J�The���k��rey�ingredien�t�that�go�S�es�in�to�pro�ving�nite�dimensionalit�y�is�the�follo�wing�result,��1whic�h���can��b�S�e�obtained�b��ry�con�tour�in�tegration:������Let��e�f��Q�2�UR�M���̽k�����and�let��D���=��f�z��5�2��C��:��I��m�(�z���)��>��0�;����j��z��j��1�;����j��R�Je�(�z���)��j��1�=�2�g��e�b�S�e�the�fundamen��rtal���domain��for��S���L���̺2����(�Z�).�8�Then��s�������.������X���"%����d�p�2�D�<r�[1��������ō��p��1���C��Qm�fe
9|���e���̽p������߰��or�S�d���̽p���]�(�f�G��)�UR=������ō�&��k�����Qm�fe�����12������H��where������t�j�e���̽p����=������ō���1�����Qm�fe����2�������#�f�
��n�2�UR�S���L���̺2����(�Z�)�:��
��p��=��p�g��=������ō���1�����Qm�fe����2�����#�Aut�(�E���̽p���]�)���p�Here,�j[the��latter�equalit��ry�follo�ws�from�the�observ��X�ation�that�the�category�of�elliptic�curv�es���o��rv�er�f��C��with�isogenies�is�the�same�as�the�category�of�lattices�in��C��upto�homothet��ry�with�maps���b�S�eing��m��rultiplication�b�y�elemets�of��C�.��zOne�can�sho�w�that�the�in�v�ertible�maps�that�preserv�e���the��Ulattice��Z����Z�p��U�are�in�one-to-one�corresp�S�ondence�with�the�set��f�
��n�2�UR�S���L���̺2����(�Z�)�:��
��p��=��p�g��U�and���hence��the�latter�equalit��ry��V.������In��particular,��-7���߈�e���̽p����=������UR�8��	��UR>����UR<����UR>����UR:�����N����8�2����!�if�꨼p�UR�=��i�����ȍ��8�3����!�if�꨼p�UR�=��������3������������p���
�������z�5S�	l��1������������8�1����!�otherwise������������9����
l���K����b�{�K������{����Using�2�this�form��rula�and�relating�the�dimensions�of��M���̽k��	Vs�and��S���̽k��#��,�D�one�can�sho�w�that��M���̽k��	Vs�is����nite��dimensional�and�also�explicitly�calculate�its�dimension.������T��Vo�u%get�a�basis�with�F�ourier�co�S�ecien��rts�in��Q�,���rst�observ�e�that��M���̽k��	���is�generated�b�y�the���set��0of�Eisenstein�series��G���̽k��
¹for�all��k�g�.�LyAs�a�function�on�the�complex�upp�S�er�half�plane,�2�the���Eisenstein��series��G���̽k��	:�for��k��o�2�UR�Z��and��k�>�UR�1�is�giv��ren�b�y��MB�����G���̽k��#��(��W�)�UR=�����+�����X���ϧ���(�m;n�)�2�Z�����2��*��n�(0�;�0)��������ō�Y��1��A���Qm�fe4{���(�m�Ź+����n�)������k�������B�One��can�then�sho��rw�that�������G���̽k��#��(��W�)�UR=������ō���1�����Qm�fe����2�����������(1������k�g�)�+�������*��1��
���1˟����X���8獓�k�6��=1���*�����̽k�6���1���(�n�)�q��n9����n�������where�꨼q�Ë�=�UR�e����2�2��I{i����,���ҩ�is�the�Riemann�zeta�function�and�������%9����̽k��#��(�n�)�UR=���������X���������d�j�n�����d�����k��� Y���There�wNis�a�theorem�due�to�Euler�whic��rh�states�that�����(1�����k�g�)�UR=��������)��33�b��i?�k���33�6	�fe9���'���2�k�������where�wN�b���̽k����are�the�Bernoulli���n��rum�b�S�ers��dened�b��ry�the�follo�wing�p�S�o�w�er�series�expansion:��z��������ō�ʵ��x���iПQm�feE���e������x�������1������7�=��������U�1��
����u�����X���8獑UR�k�6��=0��������ō���b���̽k��#��x����2�k�����Qm�fe����%�k�g�!������VߍThe�Y6constan��rt�term�of��G���̽k��	|ȹis�th�us���b���̽k��#��=��2�k��c��,�t�whic�h�is�rational.���Th�us�the�F��Vourier�expansion�of����G���̽k��	:�has��rational�co�S�ecien��rts�and�th�us�w�e�ha�v�e�found�a�basis�with�rational�co�S�ecien�ts.��� ����Next,��let��"�V�u��b�S�e�a�subspace�of��M���̽k�����whic��rh�is�stable�under�the�action�of�all�the�Hec�k�e�op�S�erators����T���̽n���P�.�S!F��Vor�H�example,�Cobserving�that��T���̽n���(�G���̽k��#��)��w=�����̽k�6���1���(�n�)�G���̽k���,�Cw��re�H�see�that��V���=��w�C�(�G���̽k���)�is�one�suc��rh���subspace.������Let�v��T��ع=��T�(�V��p�)�=��C�-algebra�generated�b��ry�the��T���̽n���P�'s�inside��E��nd�(�V��)�=��C�-v��rector�space���generated�^�b��ry�the��T���̽n���P�'s�inside��E��nd�(�V��p�).�
-The�latter�equalit�y�of�sets�follo�ws�b�S�ecause�the�pro�duct���of�M�t��rw�o�Hec�k�e�op�S�erators�can�b�e�expressed�as�a�linear�com��rbination�of�nitely�man�y�Hec�k�e���op�S�erators.������F��Vor�꨼k��o>�UR�0,�w��re�dene�a�bilinear�map��T������V��¿!��C�꨹b��ry�����j�(�T���;���f�G��)�UR�7!��a���̺1����(�f��Q�j��T��ƹ)�:���@����Prop�osition��2���U���.��)The���induc��ffe�d�maps��T��5�!��H��Vom�(�V��;����C�)��and��V�_��!��H��Vom�(�T�;����C�)��ar��ffe�isomor-���phisms.����Pr���oof.����W��Ve��rst�sho��rw�that�the�maps�are�injectiv�e.���Injectivit��ry��of�the�second�map:��6�I���ۦf�����żf��Q�2�UR�V��¿7!��0���������ſ)�UR�a���̺1����(�f��Q�j��T��ƹ)�=�0�꨿8�T���2��T��������ſ)�UR�a���̺1����(�f��Q�j��T���̽n���P�)�=�0�꨿8�n��������ſ)�UR�a���̽n���P�(�f�G��)�=�0�꨿8�n����1��������ſ)�UR�f��2��is��constan��rt���������ſ)�UR�f��Q�=�0���if�����k��o>��0����������10����z��K����b�{�K������{��Injectivit��ry��of�the�rst�map:��Auۍ���ff������T���2�UR�T��7!��0����������)�UR�a���̺1����(�f��Q�j��T��ƹ)�=�0�꨿8�f��2��V���������)�UR�a���̺1����((�f��Q�j��T���̽n���P�)��j��T��ƹ)�=�0�꨿8�f��2��V���������)�UR�a���̺1����((�f��Q�j��T��ƹ)��j��T���̽n���P�)�=�0�꨿8�f��2��V���������)�UR�a���̽n���P�(�f��Q�j��T��ƹ)�=�0�꨿8�n�>��0�;����8�f��2��V���������)�UR�f��Q�j��T���=�0�꨿8�k��o>��0�;����8�f��2��V���������)�UR�T���=�0�����B �����In��qthe�fourth�line,��|�f��p�is�replaced�b��ry��f��Q�j�UR�T���̽n���P�.�$�Next�observ�e�that��V��p�,��|b�S�eing�a�subspace�of��M���̽k��#��,����is��� nite�dimensional.� �Hence�w��re�ha�v�e�from�the�injectivities�of�b�S�oth�the�ab�o��rv�e���maps�that�eac��rh���map��is�actually�an�isomorphism.��������A��map�� �UR�:��M��6�!��N�Fҹof��T�-mo�S�dules�is�said�to�b�e��T�-equiv��X�arian��rt�if�� �(�T���m�)�UR=��T� �(�m�)��8�m��2��M�@�.��\ ����Prop�osition��3���U���.�'�The�wTisomorphisms��T�����P���UR����԰���n:�=��������H��Vom�(�V�� ;����C�)��and��V�����P����¿���԰��� ��=������k�H�om�(�T�;����C�)��as�de ne��ffd�ab�ove���ar��ffe�35�T�-e�quivariant.����Pr���oof.��x�Consider�C�the� rst�map.� C�Here�is�the��T�-mo�S�dule�structure�on��H��Vom�(�V�� ;����C�).�Giv��ren���� ��v�2�Q=�H��Vom�(�V�� ;����C�),�_�i.e.��� ��:��V��!��C�,�_�de ne��T��� ��:��V��!��C��b��ry�(�T��� �n9�)(�f�G��)�=�� ��(�f��<�j��T��ƹ).��Let�� ����denote���the�map��T���!��H��Vom�(�V�� ;����C�).�Z�Then���giv��ren��T���Ɵ��2�0��� �2���T��and��T�,Կ2��T�,�� w��re�ha�v�e�to�sho�w�that���� ��O�(�T��ƹ(�T�����2�0��o��))��=��T��ƹ(� ��(�T�����2�0��o��)).��^Let��}�f���2���V��p�.�No��rw�(� ��O�(�T���T�����2�0��o��))(�f�G��)��=��a���̺1����(�f���j��T���T�����2�0���),��rwhile��}(�T��(� ��OT�����2�0���))(�f�G��)��=���� ��O�(�T���Ɵ��2�0��o��)(�f�^��j���T��ƹ)�=��a���̺1����((�f��j��T��ƹ)��j��T�����2�0��o��)�=��a���̺1����(�f�^��j��T�T�����2�0��o��).��<Th��rus�\q(� ��O�(�T�T�����2�0���))(�f�G��)��=�(�T��(� ��O�(�T�����2�0��o��)))(�f�G��)�\q�8�f�^��2��V����and��w��re�are�done.���Next���consider�the�second�map.�s�W��Ve�de ne�the��T�-mo�S�dule�structure�on��H��Vom�(�T�;����C�).�Giv��ren���� ����2��H��Vom�(�T�;����C�),���i.e.� �$���:��T��!��C�,�dene�[i�T�����:��T��!��C��b��ry�(�T����)(�T�����2�0��o��)���=���(�T���T�����2�0���).�	�$Let�[i� ����denote���the�map��V�M�!���H��Vom�(�T�;����C�).���Then�giv��ren��f���2��V�Sd�and��T�Rտ2��T�,��w��re�ha�v�e�to�sho�w�that���� ���(�T��ƹ(�f�G��))�UR=��T��(� ���(�f�G��)).�8�Let�꨼T�����2�0���Q�2��T�.�8�No��rw����w�(� ���(�T���f�G��))(�T���� ��0��o��)�UR=��a���̺1����((�f��Q�j��T��ƹ)��j��T���� ��0��o��)�=��a���̺1����(�f��Q�j��T�T���� ��0��o��)�;����while�������-(�T��ƹ(� ��f�G��))(�T���� ��0��o��)�UR=�� ��(�f�G��)(�T���T���� ��0��o��)�UR=��a���̺1����(�f��Q�j��T���T���� ��0���)�:��r���Th��rus����� }(� ���(�T���f�G��))(�T���� ��0��o��)�UR=�(�T��ƹ(� �f�G��))(�T���� ��0��o��)�꨿8�T���� ��0���Q�2�UR�T����and��w��re�are�done.��\ ����De nition��3���K���.�d�A��2n�.�element��f�v��of��M���̽k�� R]�is�said�to�b��ffe�an�eigenform�if�it�is�an�eigenfunction�for���al���l�35the�He��ffcke�op�er�ators.���i.e.�fi�f��Q�j�UR�T���̽n�����=�����̽n���P�f�{4�for�35some�����̽n���2�UR�C��8�n����1�.����Let�Q��f����is�an�eigenform�with�eigen��rv��X�alues�����̽n���P�.� n�Then��a���̽n���(�f�G��)�l�=��a���̺1����(�f����j��T���̽n���)�=��a���̺1����(����̽n���f�G��)�=�������̽n���P�a���̺1����(�f�G��)�꨿8�n�UR���1.���Th��rus��if��a���̺1����(�f�G��)�UR=�0,��then��a���̽n���P�(�f��)�UR=�0�꨿8�n����1.���If���k��o>�UR�0�then�this�implies��f��Q�=�0.��Hence�if��k��o>��0,���then�if��f��Q�6�=�0,���w��re�can�normalize��f���to�����Fu�� �#�1��� �콉feB ��'��a�� q�1��*��(�f��Ǻ)�����0^�f�G��.��� ����11���� ����K����b�{�K������{����De nition��4���K���.�fiA��2n�35eigenform��f�{4�is�said�to�b��ffe�normalize�d�if��a���̺1����(�f�G��)�UR=�1�.�������If��F�f�#E�is�a�normalized�eigenform�with�eigen��rv��X�alues�����̽n���P�,��Zthen��a���̽n���(�f�G��)�UR=�����̽n�� ���and��F�f��Q�j��T���̽n�����=�����̽n���f��Q�=�����a���̽n���P�(�f�G��)�f��.������Again,��2let�iT� �|�denote�the�map��V��¿!�UR�H��Vom�(�T�;����C�)�induced�b��ry�the�bilinear�pairing�men�tioned���earlier.�8�Then��if��f��Q�2�UR�V��p�,�w��re�ha�v�e�the�map�� ���(�f�G��)�UR:��T��!��C�.������Prop�osition��4���U���.�FL��ffet��9�f�8�b�e�an�eigenform.�FThen��f�8�is�normalize�d��,�� ���(�f�G��)��is�a�ring�homomor-���phism.����Pr���oof.����If� �f��@�=��A0�then�the�statemen��rt�is�trivial.��So�assume��f��6�=��A0.��Then�as�discussed�ab�S�o��rv�e,����a���̺1����(�f�G��)�UR�6�=�0.�8�Also��recall�from�the�same�discussion�that�if��f��Q�6�=�UR0�is�an�eigenform,�then��! ;����f��Q�j�UR�T���̽n�����=������ō����a���̽n���P�(�f�G��)�����Qm�fe ��  ��t&�a���̺1����(�f�G��)�����!�8�f���:��!;>��F��Vor��ease�of�notation,�let�� ���̽f�� aǹdenote�the�map�� ���(�f�G��).�8�So��>����ۦf����Z!� ���̽f��w �(�T���̽n���P�T���̽m��Ĺ)��������Z!=�UR�a���̺1����(�f��Q�j��T���̽n���P�T���̽m��Ĺ)�������Z!=�UR�a���̺1����((�f��Q�j��T���̽n���P�)�T���̽m��Ĺ)�������Z!=�UR�a���̽m��Ĺ(�f��Q�j��T���̽n���P�)�������Z!=�UR�a���̽m��Ĺ((�a���̽n���P�(�f�G��)�=a���̺1����(�f��))�f��)�������Z!=�UR�a���̽m��Ĺ(�f�G��)�a���̽n���P�(�f��)�=a���̺1����(�f��)�:�����=�Q��W��Ve��ha��rv�e����e�Լ ���̽f��w �(�T���̽n���P�)� ���̽f���(�T���̽m��Ĺ)�UR=��a���̺1����(�f��Q�j��T���̽n���)�a���̺1����(�f��Q�j��T���̽m��Ĺ)�=��a���̽n���(�f�G��)�a���̽m��Ĺ(�f��)�:����The��follo��rwing�are�equiv��X�alen�t:��#UR�����&e������ ���̽f�� aǹis��a�ring�homomorphism,���������� ���̽f��w �(�T���̽n���P�T���̽m��Ĺ)�UR=�� ���̽f���(�T���̽n���P�)� ���̽f���(�T���̽m��Ĺ)�꨿8�T���̽n���;���T���̽m���,���������a���̺1����(�f�G��)�UR=�1,��and���������f�2��is��normalized.�������The�� rst�implication�follo��rws�b�S�ecause��T��is�generated�b�y�the��T���̽n���P�'s.��(V��Jan��u�uary�z�29,�1996��b#��Scrib�S�e:�8�J�� anos��Csirik,��<[email protected]>�������T��Vo�S�da��ry��mw�e'll�consider�questions�of�rationalit�y�and�in�tegralit�y��V.� /.(References:� 2iSerre:��A����Course��in�A��2rithmetic��s�and�Lang:��v�Intr��ffo�duction��to�Mo��ffdular�F���orms�.)� �BLet��s�S�<;�=��d�S���̽k�� ��b�S�e�the���space��of�cusp�forms�of�w��reigh�t�꨼k�g �.�8�Let�����d�S��׹(�Q�)�UR=��S���̽k���:�\����Q�[[�q�n9�]]��� ����12���� ����K����b�{�K������{��and�������?�S��׹(�Q�)�UR���S��(�Z�)�=��S���̽k���:�\����Z�[[�q�n9�]]�:��ٍ�The��Ffollo��rwing�fact�is�easy�to�pro�v�e�using�explicit�form�ul:�� �S���̽k�� �عhas�a��C�-basis�consisting�of���forms��with�in��rtegral�co�S�ecien�ts�(see�Victor�Miller's�construction�b�S�elo�w).������Recall��that�for�all�ev��ren��k��o��UR�4,�there�is�an�Eisenstein�series��".Y���I5�G���̽k��x�=������ō�����b���̽k������Qm�fewӟ  ���&�2�k������3�+�������m �1�� ���t*�����X��� ҁ�����n�=1������fZ���r�0�� ���[email protected]������ os����X��� ���� �"�d�j�n���-�:�d��� ��k�6���1����fZ���1�� ��A����N�I�q��n9�� ��n����;��"����whic��rh��is�a�mo�S�dular�form�of�w�eigh�t��k�g �.�8�Renormalize�this�to�obtain�� �ݍ��࠼E���̽k��x�=������ō��2�k�����Qm�fewӟ  ���b���̽k�������3�����G���̽k���=�UR1�+������������:�� ���The�� rst�few�Bernoulli�n��rum�b�S�ers��of�ev��ren�p�ositiv��re�index�are��b���̺2��V�=�UR1�=�6,����b���̺4���=���1�=�30,����b���̺6���=�1�=�42,����b���̺8�� ��=�A���1�=�30,�9�b���̺10��A��=�5�=�66,��b���̺12��A��=���691�=�2730.� _�The��Ofact�that�the� rst�four�of�these�ha��rv�e���n��rumerator��1�is�closely�related�to�the�arithmetic�of�cyclotomic� elds.������The��hmo�S�dular�forms��E���̺4�� ul�and��E���̺6���ha��rv�e��h�q�n9�-expansions�with�constan��rt�terms�equal�to�1,��and���all�X�co�S�ecien��rts�in��Z�.��The�functions��E����2����a��RA��4�����E����2����b��RA��6��� /ƹwith�4�a����+�6�b�ֹ=��k���form�X�a�basis�for��M���̽k��#��.�It�is�easy���to�see�that�they�are�mo�S�dular�forms.�� F��Vrom�the�form��rula�that�the�(w�eigh�ted)�n�um�b�S�er�of�zeros���of���an��ry�mo�S�dular�form�of�w�eigh�t��k�T�is��k�g =�12,��w�e�deduce�that��E���̺4���ֹhas�a�simple�zero�at���,��and��E���̺6�����hasn't�~�got�a�zero�at���.��Hence��E����2����a��RA��4�����E����2����b��RA��6���U��has�a�simple�zero�of�order��a��at����so�these�expressions�are���linearly��:indep�S�enden��rt�o�v�er��C�.�)�T��Vo�sho�w�that�they�span��M���̽k��#��,���consider�the�follo�wing�mo�S�dular���form��of�w��reigh�t��12:��������UR=�(�E��� �����3���ڍ4��� ÿ����E��� �����2���ڍ6���t�)�=�1728�:����Here���the�co�S�ecien��rt�of��q�. �is�a�simple�n�um�b�S�er:��61.��a�has�a�simple�zero�at��1�.�Since�a�cusp���form�R of�w��reigh�t�R �k��&�has��k�g =�12�zeros,�k�it�follo��rws�that�(since�the�w�eigh�ting��e�����1�� R�is�1),�k�that��do�S�es���not��/v��X�anish�an��rywhere�on��H�.��Therefore��S���̽k�6��+12���d�=�UR�囿��M���̽k��#��.�Since��/�E���̽k���(�i�1�)�UR=�1��6�=�0,��zit�follo��rws�that����M���̽k��Ð�=����E���̽k�����Ȇ�C����S���̽k���=����E���̽k������C��������M���̽k�6���12��@�.��|Hence���dim���ͼM���̽k���=����dim��5D�M���̽k�6���12����+�1.��|Again��using�the���fact�� that��f��:�2��;�M���̽k�� װ�has��k�g =�12�zeros,��|w��re�quic�kly�deduce�that�the�dimensions�of��M���̺0����,��|�M���̺2���,��M���̺4���,����M���̺6����,�3ϼM���̺8���,��M���̺10�� %5�are�%-1,0,1,1,1,1�resp�S�ectiv��rely��V.��p(e.g.,�for��k� �=���4�an��ry�mo�S�dular�form�m�ust�ha�v�e�just���a���simple�zero�at���.�*�So�for�an��ry��f����2�o��M���̺4����,�� it�is�the�case�that��f�G��(� �W �)�����f��(�i�1�)�E���̺4����(� �W �)���v��X�anishes�at���� � +�=���i�1�,�and�hence�is�iden��rtically�zero.���So��E���̺4����spans��M���̺4����.)�Th��rus�w�e�ha�v�e�determined��dim���)�M���̽k�����for�\�all��k�~T��7�0,�y and�it�is�easy�to�see�that�this�n��rum�b�S�er�\�is�equal�to�the�n��rum�b�er�\�of�solutions�to���4�a���+�6�b��q�=��k�y˹for���a;���b����0.���Hence�the��E����2����a��RA��4�����E����2����b��RA��6���项span��M���̽k�� [email protected]�and�therefore�w��re�ha�v�e�pro�v�ed�that�they���form��a�basis.������The�ffollo��rwing�construction�comes�from�the� rst�page�of�Victor�Miller's�thesis.��:Let��d�'s�=����dim����H����C�� )3�S���̽k��#��.�0(Then�Ёthere�exist��f���̺1����;����:�:�:��ʜ;���f���̽d��4��2�UR�S��׹(�Z�)�suc��rh�that��a���̽i��dڹ(�f���̽j��f �)�=�����̽ij�� e�for�1����i;���j�%���d�.�0(T��Vo�sho��rw���this,��nrecall���that��E���̺4�� c�2�V_�M���̺4�� A��and��E���̺6���2�V_�M���̺6�� A��ha��rv�e����q�n9�-expansions�with�co�S�ecien��rts�in��Z�.�����2��S���̺12�����has��constan��rt�co�S�ecien�t�0,��and�the�co�S�ecien�t�of��q�|�is�1.���Also�����2��S���̺12�� �(�Z�),��as��can�b�S�e�seen�for���example��from�the�form��rula��� ������UR=��q�������0��1�� ��� �����Y��� ҁ��n7�n�=1���s�(1������q��n9�� ��n����)��� ��24�� �:��� �����13�������K����b�{�K������{��No��rw�Q pic�k��a;���b��S���0�so�that�14����4�a����+�6�b��S���k� g�(�mo�S�d���B12),���with��a��=��b��=�0�when��k� p���0����(�mo�S�d���B12).�3ENote���that�then�12�d��R�+�6�a��+�4�b�UR�=��k�@��b��ry���our�previous�result�on�the�dimension�of��M���̽k�����(and�|9the�fact�that�the�dimension�of��S���̽k���˹is�one�less�than�that�for��k��o��UR�12).�Hence�the�functions��ލ���f�g���̽j���\�=�UR��� ��j��f �E����ߍ���2(�d��j�v�)+�a���g��6���&�E��� �����b���ڍ�4����w���for���1�UR���j�%���d��will�b�S�e�cusp�forms�of�w��reigh�t��k�g �.�3�By�our�previous�remarks�on�the�co�S�ecien�ts�of���,�꨼E���̺6����,��E���̺4���,�w��re�ha�v�e��g���̽j���\�2�UR�S���̽k��#��(�Z�)�and�����]�a���̽i��dڹ(�g���̽j��f �)�UR=�����̽ij����Q��for�y �i�UR���j��ӹ.�A�ystraigh��rtforw�ard�elimination�no�w�yields�the��f���̺1����;����:�:�:��ʜ;���f���̽d��Xǹwith�the�stated�prop�S�erties.���It�7�is�clear�that�these��f���̺1����;����:�:�:��ʜ;���f���̽d�� e�are�linearly�indep�S�enden��rt�o�v�er��C�,�Khence�they�form�a�basis�of����S���̽k��#��.������If���y��rou�tak�e��T���̺1����;����:�:�:��ʜ;���T���̽d�� ���2����T��=��T�(�S���̽k��#��),��they���are�also�linearly�indep�S�enden�t:���for�giv�en�an�y���linear��relation���<����������U�d�� �����D�����X��� ��)~�i�=1����C �c���̽i��dڼT���̽i���,�=�UR0�;��};��apply��this�to��f���̽j��P��and�lo�S�ok�at�the� rst�co�ecien��rt�� ����l 0�UR=��a���̺1�������f\� �����f���̽j���������f� ����f ����f ����f ����f ����������n�d�� ��� �]�����X��� � ��i�=1���-$�c���̽i��dڼT���̽i����.���f\�!���M��=�������^c�d��
��������X�������i�=1�����c���̽i��dڼa���̽i���(�f���̽j��f
�)�UR=�������^c�d��
��������X�������i�=1����c���̽i��dڼ���̽ij��
�6�=��c���̽j��f
�;��A���hence��the�linear�relation�giv��ren�in�the�rst�place�w�as�trivial,��]so��T���̺1����;����:�:�:��ʜ;���T���̽d��	���form�a�basis�for����T�(�S���̽k��#��),��since�they�are�linearly�indep�S�enden��rt,�and��dim�������C��!��T�UR�=��dim���ꚟ���C�� ~��V��¹=��d�.������Let�꨿R�UR�=��Z�[��:���:�:��ʞT���̽n����N�:���:�:��r�]�����End��b(�S���̽k��#��).��������Claim��1���������oݿR�UR�=����������d��
��������M�����i�=1���q��Z�T���̽i��dڼ:��!O卑���Pr���oof.����Since��the��T���̽i��L۹form�a�basis�of��T�,��w��re�ha�v�e��T���̽n�����=��UR�����P����*���
㍽d���̍�
�i�=1���$�c���̽n��8:�i����¼T���̽i��L۹with��c���̽n��8:�i��� *�2�UR�C�.�7�W��Ve�need���to��c��rhec�k�that��c���̽n��8:�i��� *�2�UR�Z�.�8�With�the��f���̽j��P��as�ab�S�o�v�e,�consider��6������ʘ������a���̽n���P�(�f���̽j��f �)������=�UR�a���̺1����(�f���̽j��f �j�T���̽n���P�)���Y�������=�UR�a���̺1�������f��� ��f���̽j���������f� ����f ����f ���� �]����P����*���I��d���̍�I�i�=1���$��c���̽n��8:�i����¼T���̽i����=�K��f������c��������=��UR�����P����*���
㍽d���̍�
�i�=1���$�c���̽n��8:�i����¼a���̺1����(�f���̽j��f �j�T���̽i��dڹ)����.������=��UR�����P����*��� ㍽d���̍� �i�=1���$�c���̽n��8:�i����¼a���̽i��dڹ(�f���̽j��f
�)���������=��UR�����P����*���
㍽d���̍�
�i�=1���$�c���̽n��8:�i����¼���̽ij����)�%�=�UR�c���̽n��8:�j��� �+�:�����5�C��Hence�꨼c���̽n��8:�i��� *�2�UR�Z�.��o%�&���� msam10�������R���is�called�the��inte��ffgr�al�� He�cke�algebr�a�.�+�It���is�a� nite��Z�-mo�S�dule�of�rank��d�.�W��Ve�still�ha��rv�e���(from��the�last�lecture)�a�pairing����������S��׹(�Z�)�����R�������t!�������v�Z����������}�(�f���;���T��ƹ)�������t�7!�������v�a���̺1����(�f�G��j�T��ƹ)�:����� �����14��������K����b�{�K������{��No��rw�/�S��׹(�Z�)�~��,���!���Hom�� ��(�R�;����Z�)�����P�������԰�����=�����SQ�Z����2�d�� ��b�y�/�the�argumen�t�giv�en�b�S�efore.� �Therefore��S��׹(�Z�)�is�a�free�����Z�-mo�S�dule��of� nite�rank.�8�But�it�also�con��rtains�the��f���̽i��dڹ,�so��S��׹(�Z�)�����P���UR����԰���n:�=��������Z����2�d��ߨ�.������What��is��S��׹(�Z�)�as�an��R�-mo�S�dule?�������Exercise��1���B��The�35map��S��׹(�Z�)�UR�,���!���Hom����(�R�;����Z�)�35�is�in�fact�an�isomorphism�of��T�-mo��ffdules.�������Hint.� 3�The��cok��rernel�is�a�torsion�(in�fact� nite)�group.��<So�if�w�e�sho�w�it�torsion�free,� �w�e����are��done.������Theorem��3���E���The�35�T���̽n�� ۅ�ar��ffe�al���l�diagonalizable�on��S���̽k��#��.������S���̽n�� ���supp�S�orts��a�Hermitian�non-degenerate�inner�pro�duct,�the�P��retersson�inner�pro�duct�������$(�f���;���g�n9�)�UR�7!�h�f�;���g�n9�i�2��C�:����W��Ve��ha��rv�e��h�f���;���f�G��i�UR��0,�with�equalit��ry�i��f��Q�=�0.�8�F��Vurthermore�������h�f�G��j�T���̽n���P�;���g�n9�i�UR�=��h�f���;�g�n9�j�T���̽n���P�i�;����i.e.,�꨼T���̽n��	���is�self-adjoin��rt�with�resp�S�ect�to�the�giv�en�inner�pro�S�duct.������An�5op�S�erator��T��۹is��normal��if�it�comm��rutes�with��T���Ɵ��2���	�߹(whic�h�denotes�its�Hermitian�transp�S�ose).���Normal�R5op�S�erators�are�diagonalizable�(for�a�pro�of,�p�refer�to�Math�H110).�In�our�case,��T����2�������RA��n���	��=�UR�T���̽n���P�,���so�M?this�fact�applies.�hIt�is�also�true�(same�pro�S�of��8)�that�a�comm��ruting�family�of�semi-simple�(i.e.,���diagonalizable)��op�S�erators�is�sim��rultaneously�diagonalizable.�������Pr���oof.����Put��together�the�ab�S�o��rv�e��facts.���g��������W��Ve�|�can�also�pro��rv�e�|�that�the�eigen��rv��X�alues�are�real.�)This�dep�S�ends�on�the�follo�wing�tric�k.�)F��Vor����f��Q�6�=�UR0��consider����E�c�a���̽n���P�h�f���;���f�G��i�UR�=��h�a���̽n���f���;���f�G��i��=��h�f��j�T���̽n���P�;���f��i��=��h�f���;���f��j�T���̽n���P�i��=��h�f���;���a���̽n���f�G��i��=���N�����a���̽n����(��h�f���;���f��i�:���a���̽n�����2�UR�R�꨹no��rw�follo�ws�since��f��Q�6�=�UR0�implies��h�f���;���f�G��i�6�=�0.�������Exercise��2���B��The�35�a���̽n��	ۅ�ar��ffe�total���ly�r�e�al�algebr�aic�inte�gers.�������Hint.�
���The�7ispace��S���̽k��	Z��is�stable�under�the�action�of��Aut��.(�C�)�\on�the�co�S�ecien��rts".�$Giv�en���a��cusp�form��� ���lK�f��Q�=���������1�� ��� ԟ����X��� ҁ��UR�n�=1���Z �c���̽n���P�q��n9�� ��n��� R���and���some���Ë�2��UR�Aut��=(�C�),���de ne�����2���'��f��Q�=��UR�����P����*��� ��1���̍� ㍽n�=1��� hW��n9�(�c���̽n���P�)�q�����2�n����.�QThis���function�is�in��S���̽k��#��,�since��S���̽k�����has�a�basis���in�꨼S��׹(�Q�),�whic��rh�is� xed�b�y���n9�.�8�Then��f�2��is�an�eigenform�i �����2������f��is�an�eigenform.������W��Ve'll�yuse�a�lame�de nition�of�the��Petersson��ginner�pr��ffo�duct�y�for�this�section.� �Let��z��U�=����x��@�+��iy���2���H�.��\Then�['w��re�ha�v�e�a�v�olume�form��y��n9���2��2��� ʵ�dx���dy��+2�whic�h�is�in�v��X�arian�t�under��GL������+�����2��� x�(�R�)�(the��� ����15����Ĭ��K����b�{�K������{��subgroup���of�the�general�linear�group�of�matrices�of�p�S�ositiv��re�determinan�t).�0�T��Vo�pro�v�e�this,����note��that��dz�3��^����d����9����z��� S��=�UR(�dx��+��i���dy�n9�)��^��(�dx����i�dy�n9�)�UR=���2�i�(�dx����^��dy��),��and�hence�� ������߼dx���dy�Ë�=�UR�dx����^��dy��=������ō�����1�����Qm�fe5S�  ����2�i������ dz�3��^��d����9����z����';G��Then��for�an��ry�� �h�=���UR��f\� �����d���?��a��� � b��������7c��� j�d�����+���f\�!���6���2�UR�GL������+�����2��� x�(�R�)�w�e�can�consider�the�usual�action��* E����o� �h�:�UR�z��5�7!������ō����az�3��+����b�����Qm�fe ���  ��-cz�3��+����d�����(��=������ō���(�az�3��+����b�)(�c����9����z����ݹ+��d�)�����Qm�feQ���  ��>��j�cz�3��+����d�j������2�������!�ߍ�where��othe�imaginary�part�of�the�result�is��j�cz�� �+�H)�d�j����2��2�� \|�(�ad����bc�)�Im��Ǽz��5�=�UR�y�n9�j�cz��+��d�j����2��2��� \|�det���8(� ���)�:��o�As�for���di eren��rtials,��U]��Z�d�������f\� �������ō� ݼaz�3��+����b�� ݟQm�fe ���  ��-cz�3��+����d������+���f\�!���72�=������ō����a�(�cz�3��+����d�)����dz����(�az��+��b�)�c���dz�����Qm�fe��G�  ��- ��(�cz�3��+����d�)������2��������Q�=������ō�ˉ�ad������bc�����Qm�fe-�*�  ��(�cz�3��+����d�)������2������5?�dz���^��hence��under�application�of�� ���,��dz�3��^����d����9����z��� �ݹtak��res�on�a�factor�of��#R��������ō�����det���/�(� ���)�����Qm�fe-�*�  �(�cz�3��+����d�)������2�����������ō���ȃ�det���?(� ���)����t�Qm�fe-�*�  �(�c����9����z����ݹ+����d�)������2�������#�=���UR��f\� �������ō��wҹdet��!Ɏ(� ���)�� s1�Qm�fe+ N�  ��j�cz�3��+����d�j������2�������8����f\�!����@�^��*��2��G\�:��"�ۍ�This�� nally�pro��rv�es��that�the�di eren��rtial��y��n9���2��2��� ʵ�dx���dy��*���is�in�v��X�arian�t�under�the�action�of��GL������+�����2��� x�(�R�).������The��form��rula�for�the��Petersson�35inner�pr��ffo�duct�꨹is�� �㍒����h�f���;���g�n9�i�UR�=����㇫Z���㌟�z�S�r}L�� q�2��*��(�Z�)�n�0\��%eufm8�H����1���f���7��f�G��(�z���)����щfe0M� 3/��g��(�z��)���0M�y���� ��k����˟�f���������ō�y� �dx���dy��y� �Qm�fe 3�  ���y�������2�������ނ�:�� �΍���This�5ycould�b�S�e�considered�either�an�in��rtegral�o�v�er�the�fundamen�tal�region�or,�H-noting�that���the�k�in��rtegrand�is�in�v��X�arian�t�under��S���L���̺2����(�Z�),���an�in�tegral�o�v�er�the�quotien�t�space.� ��One�then���c��rhec�ks���that�for��f���;���g�G �cusp�forms,��rthe�in��rtegral�will�con�v�erge,��rsince�they�go�do�wn�exp�S�onen�tially���as�꨼z�s��tends�to�in nit��ry��V.�8�It�is�then�clear�that�this�inner�pro�S�duct�is�Hermitian.������It��is�not�immediately�clear�that�the�Hec��rk�e��op�S�erators�are�self-adjoin��rt.��(V��Jan��u�uary�z�31,�1996��b#��Scrib�S�e:�8�William��Stein,��<[email protected]>��������Mo�s3dular�ffCurv���es��� �����16����ў��K����b�{�K������{���0.0.1��1�Cusp�ffF���forms��@��Recall�$�that�if��N�e��is�a�p�S�ositiv��re�in�teger�w�e�dene�the�congruence�subgroups�(�N�@�)��?������̺1����(�N��)����������̺0����(�N�@�)��b��ry��@�����썍�A�3���̺0����(�N�@�)���k��=�UR�f����f\� �����d���ꬼa�����b�������
x�c����d�����(0���f\�!���3p��2���SL���3ݟ�̺2����(�Z�)�:��c����0���(�mo�S�d���B�N�@�)�g������A�3����̺1����(�N�@�)���k��=�UR�f����f\� �����d���ꬼa�����b�������
x�c����d�����(0���f\�!���3p��2���SL���3ݟ�̺2����(�Z�)�:��a����d����1�;���c����0���(�mo�S�d���B�N�@�)�g������D5�(�N�@�)���k��=�UR�f����f\� �����d���ꬼa�����b�������
x�c����d�����(0���f\�!���3p��2���SL���3ݟ�̺2����(�Z�)�:�����f\� �����d���?��a��� �b��������7c��� j�d�����+���f\�!���6��������(�������d����1����0��������0����1�������'�)����88�(�mo�S�d���B�N�@�)�g�����@����Let���b�S�e�one�of�the�ab�o��rv�e��subgroups.��One�can�giv��re�a�construction�of�the�space��S���̽k��#��()���of�fcusp�forms�of�w��reigh�t�f�k�|��for�the�action�of��using�the�language�of�algebraic�geometry��V.��Let����X���̺��
���=����n�H����2���	~Q�b�S�e��Mthe�upp�er�half�plane�(union�the�cusps)�mo�dulo�the�action�of�.���Then��X���̺�����can�b�S�e�giv��ren�the�structure�of�Riemann�surface.�+�F��Vurthermore,�ʎ�S���̺2����()�UR=��H���V���2�0���Z�(�X���̺����;����
����2�1���)�where�
����2�1�����is���the�sheaf�of�dieren��rtial�1-forms�on��X���̺����.��This�w�orks�since�an�elemen�t�of��H���V���2�0���Z�(�X���̺����;����
����2�1����)�is�a���dieren��rtial��form��f�G��(�z���)�dz��,��.holomorphic��on��H��and�the�cusps,�whic��rh�is�in�v��X�arian�t�with�resp�S�ect�to��	�the��action�of�.�8�If��
��n�=���UR��f\� �����d���?��a��� �b��������7c��� j�d�����+���f\�!���6���2�UR��then��#D���\Լd�(�
���(�z���))�=dz��5�=�UR(�cz�3��+����d�)������2������so�����7O�f�G��(�
���(�z���))�d�(�
��(�z��))�UR=��f�G��(�z��)�dz����i�꨼f�2��satises�the�mo�S�dular�condition�����o߼f�G��(�
���(�z���))�UR=�(�cz�3��+����d�)�����2����f��(�z���)�:������There��is�a�similiar�construction�when��k��o>�UR�2.��"ʫ���0.0.2��1�Mo�s3dular�ffCurv���es��@���SL���
ދ��̺2�����(�Z�)�n�H��Ϲparametrizes�isomorphism�classes�of�elliptic�curv��res.��TThe�other�congruence�sub-���groups��Aalso�giv��re�rise�to�similiar�parametrizations.�h���̺0����(�N�@�)�n�H��parametrizes�pairs�(�E��;���C�ܞ�)�where����E��1�is��an�elliptic�curv��re�and��C�Ҹ�is�a�cyclic�subgroup�of�order��N�@�,�8�and����̺1����(�N��)�n�H��parametrizes���pairs�%�(�E��;���P��ƹ)�where��E���is�an�elliptic�curv��re�and��P�ǒ�is�a�p�S�oin�t�of�exact�order��N�@�.��LNote�that�one���can��]also�giv��re�a�p�S�oin�t�of�exact�order��N��A�b�y�giving�an�injection��Z�=��X�N�@��Z�UR�,���!��E���[�N��]��]or�equiv��X�alen��rtly���an�}injection�����̽N��g��,���!�No�E���[�N�@�]�where�����̽N���G�denotes�the��N��th�ro�S�ots�of�unit��ry��V.���(�N��)�n�H��parametrizes���pairs��(�E��;����f���;���O�g�)�where��f���;���O�g��is�a�basis�for��E���[�N�@�]�����P���UR����԰���n:�=�������(�Z�=��X�N��Z�)����2�2����.������The���ab�S�o��rv�e�quotien�ts�spaces�are��mo��ffduli��7sp�ac�es��for�the��mo�duli��7pr�oblem��of�determining���equiv��X�alence��classes�of�pairs�(�E���+�extra�structure).�������17�����_��K����b�{�K������{���0.0.3��1�Classifying�ff�X�Qffcmr12�(�N���)�-structures��@����Denition��5���K���L��ffet��~�S��U�b�e�an�arbitr�ary�scheme.��EA��2n��:F
C�
cmbxti10�el���liptic���curve��E��=S��U�is�a�pr�op�er�smo�oth����curve��������fd����-F�E�����������5��j�����������@�S�����"�with�35ge��ffometric�al���ly�c�onne�cte�d�b�ers�al���l�of�genus�one,�to�gether�with�a�se�ction�\0".��9*����Lo�S�osely�F�sp�eaking,��prop�er�generalizes�the�notion�of�pro��jectiv��re,��and�smo�oth�generalizes���nonsingularit��ry��V.�؊See�ɧChapter�I�S�I�I,�ɧsection�10�of�Hartshorne's��A��2lgebr��ffaic�)�Ge�ometry�ɧ�for�the�precise���denitions.������Denition��6���K���L��ffet�X�S��/�b�e�any�scheme�and��E��=S��/�an�el���liptic�curve.�\uA�Q�(�N�@�)�-structur��Ke��on��E�=S����is�35a�gr��ffoup�homomorphism������ �'�UR�:�(�Z�=��X�N�@��Z�)�����2��V�!��E���[�N��](�S��׹)���p��whose�35image�\gener��ffates"��E���[�N�@�](�S��׹)�.������See��
Katz�and�Mazur,����A��2rithmetic���Mo��ffduli�of�El���liptic�Curves�,�1985,�Princeton��
Univ��rersit�y���Press,��esp�S�ecially�c��rhapter�3.������Dene��a�functor�from�the�category�of��Q�-sc��rhemes�to�the�category�of�sets�b�y�sending�a���sc��rheme�꨼S���to�the�set�of�isomorphism�classes�of�pairs��b���=c(�E��;����(�N�@�)�-structure��2T")���where��?�E��V�is�an�elliptic�curv��re�dened�o�v�er��S���and�isomorphisms�(preserving�the�(�N�@�)-structure)����are�y�tak��ren�o�v�er��S��׹.��An�isomorphism�preserv�es�the�(�N�@�)-structure�if�it�tak�es�the�t�w�o�distin-���guished��generators�to�the�t��rw�o��distinguished�generators�in�the�image�(in�the�correct�order).������Theorem��4���E���F���or�8�N��6��UR�4��the�functor�dene��ffd�ab�ove�is�r�epr�esentable�and�the�obje�ct�r�epr�esenting���it�35is�the�mo��ffdular�curve��X��(�N�@�)��c�orr�esp�onding�to��(�N�@�)�.������What���this�means�is�that�giv��ren�a��Q�-sc�heme��S��׹,���the�set��X��(�S��)�{�=��M�@�or��33�Q��-sc��rhemes���4��(�S��;���X��)���is���isomorphic��to�the�image�of�the�functor's�v��X�alue�on��S��׹.������There���is�a�natural�w��ra�y���to�map�a�pair�(�E��;����(�N�@�)�-structure��2T")�to�an��N��th�ro�S�ot�of�unit��ry��V.��If����P�S�;���Q�꨹are�the�distinguished�basis�of��E���[�N�@�]�w��re�send�the�pair�(�E�;����(�N�@�)�-structure��2T")�to��b���b6�e���̽N��D�(�P�S�;���Q�)�UR�2�����̽N�����where������LF�e���̽N��n��:�UR�E���[�N�@�]������E��[�N�@�]�UR�!�����̽N����p��is���the�W��Veil�pairing.��F�or�the�denition�of�this�pairing�see�c��rhapter�I�S�I�I,���section�8�of�Silv�erman's����The�đA��2rithmetic�of�El���liptic�Curves�.��The�rJW��Veil�pairing�is�bilinear,��]alternating,�non-degenerate,���galois��in��rv��X�arian�t,�and�maps�surjectiv�ely�on�to�����̽N��D�.�������18�����0��K����b�{�K������{��F��aGebruary�z�2,�1996��b#��Scrib�S�e:�$La��rwren��/Smithline,��{�<[email protected]>��(Note.� �These�w�ere�done�ab�S�out�a�mon�th�after�the����fact,��since�the�assigned�p�S�erson�dropp�ed�the�course.�8�So�they�are�terse.)��G���Earlier,��w��re��Nlo�S�ok�ed�at�V.�Miller's�construction�for�eigenforms.�p�(See,��for�instance,�Lang�X����x�4���in�the�course�references.�%-This�is�a�sp�S�ecial�miracle�for��SL������̺2��N �(�Z�).)�Ov��rer��Z�,��a�T�����P���UR����԰���n:�=��������Z����2�d���7�and��S��׹(�Z�)���are��dual,�where��d�UR�=��dim��꘼S���̽k��#��(�C�).������Here���is�Shim��rura's�explanation�(Lang�I�S�I�I����x�5,��DVI�I�I).���The�Hec�k�e�op�S�erator��T���̽n�� 1�maps��S���̽k�� �P�to���itself.�8�Let�꨼A�UR���C��b�S�e�a�subring�and�� 荒�7�T���̽A�� 36�=�UR�A�[�T���̽n�����:��n�>��0]�����End���b����C�� ���S���̽k��#��:����Denote��b��ry��T�,��T�����Z��}+�.�8�There�is�a�natural�tensor�pro�S�duct��T���̽A�� ��� ���̽A���C�UR���T�����C����.�������There��}is�also�a�complex�conjugation�automorphism�of��S���̽k�� ��b��ry��f���7!��~���щfe �� 3/��f�G��(�����E���� ���qi�)���${�.�EThis�map�is���conjugate��linear.���The�map����?�7!��R"�exp��E�(2��n9i�W�)�b�S�ecomes����7!��R"���щfe;�r�
3/���exp���(��2��n9i����E�������qi�)���F|�=��R"exp��E�(2��n9i�W�).���Sa��ry����f�i��=��!������P���ݼa���̽n���P�q��n9���2�n����:�b��Its�conjugate��g��ݹ=��!������P����������a���̽n�����.�q��n9���2�n����:��If�y��rou�kno�w��S���̽k��#��(�C�)�!�=��C��b�
�����Q��
�ؼS���̽k���(�Q�),���then�b�y��rou�kno�w���that��mo�S�dular�forms�can�b�e�conjugated�in�this�sense.������There��4is�an�isomorphism��T�����R��
�@�
�����R���C��������	�����g	�����]�����
�h!����y�T�����C���!�since��4the�map�is�surjectiv��re�and�the�complex���dimensions��on�eac��rh�side�are�equal.������Shim��rura��(1959)�exhibited�the�(Eic�hler-)Shim�ura�isomorphism��荒��q�S���̽k��#��(�C�)�����P���UR����԰���n:�=��������H���V����1���Z�(�X���̺����;����R�)�:����W��Ve�Hyha��rv�e��S���̽k��#��(�C�)��=��H���V���2�0���Z�(�X���̺����;����
����2�1����)�and�a�map��H���̺1���(�X���̺����;����Z�)�ꈿ��S���̽k��#��(�C�)���!��C�.�RRNo��rw,�_�H���̺1����(�X�Jg;��Z�)�����P�������԰���
�=�����?Y�Z����2�2�d������em��rb�S�eds��in��Hom���d1����C��#��(�S���̽k��#��(�C�)�;����C�)�����P���UR����԰���n:�=��������C����2�2�d��
T�as�a�lattice.������So��hw��re�ha�v�e��S���̽k��#��(�C�)����!���Hom��+(�H���̺1����(�X�Jg;����Z�)�;��C�))��hand��S���̽k���(�C�)��������	Jg���g	�����!�����1F�Hom��1��(�H���̺1����(�X�Jg;����Z�)�;��R�))��has�real���v��rector��!spaces.�tKBy�the�Shim�ura�isomorphism,�ؿthis�is�isomorphic�to��H���V���2�1���Z�(�X�Jg;����R�)������H����2���V�1��RA��p����(�;����R�),���the��parab�S�olic�cohomology�of�.������So�L�S���̽k��#��(�C�)�����P��������԰�����=�����K��H����2���V�1��RA��p����Z�(�;���V���̽k���),�defor�a�certain��d������1�dimensional�subspace��V���̽k��#��.�](Let��W��ݹ=���R����R�.��m���acts�b��ry�linear�fractional�transformation.���Let��V���̽k���ܹ=���JSym�������x�k�6���2��*��W��ƹ.�There�is�a�lattice�in��S���̽k��#��(�C�)���corresp�S�onding��to��H����2���V�1��RA��p����Z�(�;�����Sym����D���x�k�6���2��(�R�Z����2�2����))�W��Ve�ha��rv�e��an�action�of��b��ry��"b���8˼f������
��n�7!��UR��㇫Z����2��
US�
�x�(���q�0��*��)���H��㌽��q�0����#;μf�G��(��W�)�������k�6���1��W+�d��:���'����Recall�a��T���=��T�����Z��
޽�is�a�set�of�endomorphisms�of�a�lattice��L�,�Mand��T��has�nite�rank�o��rv�er��Z�.���W��Ve��ha��rv�e�the�inclusion��S���̽k��#��(�Z�)�UR�,���!��S���̽k���(�C�),��or�equiv��X�alen��rtly��V,��荍�90Hom���Q������Z��X�(�T�;����Z�)�UR�,���!���Hom����۟���Z��"L�(�T�;��C�)�=��Hom����(�T�����C����;��C�)�=��S���̽k��#��(�C�)�=��S���̽k���(�Z�)����
�����Z��	'��C�:������Here�o�is�a�nift��ry�inner�pro�S�duct�(the�P�etersson�innner�pro�S�duct)�on��S���̽k��#��(�C�).��F��Vor��f���;���g�Ë�2�UR�S���̽k���(�C�),����let��s����|�h�f���;���g�n9�i�UR�=����㇫Z���㌟�z��n�H����f�G��(��W�)�g��(���)�y������k�������ō����dx���dy�����Qm�fe3����y�������2������#d�:��������19�����Π�K����b�{�K������{��The��Hec��rk�e�op�S�erators�are�self-adjoin�t�for�(�p;���N�@�)�UR=�1:��,�����h�f�G��j�T���̽p���]�;���g�n9�i�UR�=��h�f���;�g�n9�j�T���̽p���]�i�:����Indeed,��for���h�2��UR�GL�����x����+��������2���e�(�R�),������!�h�f�G��j���;���g�n9�j���i�UR�=��h�f���;�g�n9�i�:��(	��F��aGebruary�z�5,�1996��b#��Scrib�S�e:�8�Sh��ruzo��T��Vak��X�ahashi,��<[email protected]>��P�����W��Ve���ha��rv�e�studied�actions�of��T��on��S���̽k��#��(�C�),�S���̽k���(�Q�)�and��S���̽k���(�Z�)�where��Z�=��S���L���̺2����(�Z�).�B.What�w��re���kno��rw��so�far�is�����ݱ�S���̽k��#��(�Z�)�UR�'���Hom����۟���Z��"L�(�T�;����Z�)�:���j��Also��w��re�ha�v�e�studied�the�P�eterson�pro�S�duct.�8�It�is�Hermitian,�i.e.,��,�����^�h�f�G��j�T���̽n���P�;���g�n9�i�UR�=��h�f���;�g�n9�j�T���̽n���P�i����for�꨼T���̽n�����2�UR�T��and�for��f���;���g�Ë�2��S���̽k��#��(�C�).����Note:�8�T���̽n��	���dened��on��S���̽k��#��(�C�)�preserv��res��S���̽k���(�Z�).������T��Vo�S�da��ry��w�e�study�when��UR=����̺1����(�N�@�)�or����̺0���(�N�@�)�for��N��6��UR�1.��hG��1.�32The�ffDiamond�Op�s3erator�and�the�Decomp�osition�of��S��(��
�b>

cmmi10�k��V�(��(��K�y

cmr10�1����(�N���))��!����Theorem��5���E�޹���̺1����(�N�@�)�35�is�a�normal�sub��ffgr�oup�35of�����̺0���(�N�@�)��and�we�have�����̺0���(�N�@�)�=����̺1���(�N��)�UR�'��(�Z�=n�Z�)����2����.�� �����Denition��7���K���The��+diamond�op��ffer�ator��+�<>��is�dene��ffd�as�fol���lows:��Vfor�����f^���������L��a���!�b��������c��� w�d������(����f^����4�X�2��>����̺0����(�N�@�)�,�ithe��@�map�35on��S���̽k��#��(���̺1����(�N�@�))�����N��f��Q�!�UR�f�G��j�������f^���������\p�a����b��������c����yd������%�y��f^������j��denes��an�endmorphism��<�URd�>���of��S���̽k��#��(���̺1����(�N�@�))��which�dep��ffends�only�on��d���mo�S�d��V�N��.�TZThus�we�get����an�35action��<>��of��(�Z�=n�Z�)����2����9�on��S���̽k��#��(���̺1����(�N�@�))�.������Since��(�Z�=n�Z�)����2������is�a�nite�group,�w��re�ha�v�e�the�follo�wing�decomp�S�osition�theorem:������Theorem��6���������S���̽k��#��(���̺1����(�N�@�))�UR=���������M�����	+�����qżS���̽k���(���̺0����(�N�@�)�;����)��\n��wher��ffe�35���runs�over�the�set�of�char�acters��(�Z�=n�Z�)����2���V�!�UR�C����2����9�and��S���̽k��#��(���̺0����(�N�@�)�;����)��is�dene�d�as:��,���f��S���̽k��#��(���̺0����(�N�@�)�;����)�UR=��f�f��Q�2��S���̽k���(���̺1����(�N�@�))�:��f�G��j��<�d�>�=���(�d�)�f��g�:����We�35have��S���̽k��#��(���̺0����(�N�@�)�;����)�UR=�0�35�unless���(��1)�UR=�(��1)����2�k���.��������20����P��K����b�{�K������{��2.�32The�ffHec���k�e�Op�s3erators�on��S��(��k��V�(��(��1����(�N���))�������F��Vor�=��n��d���1,��qw��re�ha�v�e�the�op�S�eration�on��S���̽k��#��(���̺1����(�N�@�))�of�the��n�th�Hec�k�e�op�S�erator��T���̽n���P�.�	1�The���follo��rwing��are�basic�prop�S�erties:�������Theorem��7���E���(1)�35�T���̽n���P�'s�c��ffommute�e�ach�other�and�with��<�URd�>�.������(2)�35�T���̽n���P�'s�pr��ffeserve��S���̽k��#��(���̺0����(�N�@�)�;����)�.������(3)�35if��(�n;���N�@�)�UR=�1�,�35then��h�f�G��j�T���̽n���P�;�g�n9�i�UR�=��h�f���;�g�n9�j��<�n�>����=���2��1��+Q��T���̽n���P�i�.������(4)�35�h�f�G��j�UR�<�d�>;���g�n9�i��=��h�f���;�g�n9�j��<�d�>�������W��1��*i2�i�.������(5)�35if��(�n;���N�@�)�UR=�1�,�35then��T���̽n��	ۅ�is�diagonalizable.������(6)�35if��(�n;���N�@�)�UR�6�=�1�,�35then��T���̽n��	ۅ�is�not�diagonalizable.������The��action�of��T���̽n��	���is�describ�S�ed�in�the�follo��rwing�theorem:������Theorem��8���E���L��ffet�35�f��Q�=��UR�����P����*���
��1���̍�
㍽n�=1��� hW�a���̽n���P�q��n9���2�n��	kۿ2�UR�S���̽k��#��(���̺0����(�N�@�)�;����)�.�fiThen������(1)�GѼf�G��j�T���̽p��Bٹ=��{|�����P����*���	��1���̍�	��n�=1��� ���a���̽pn��	ﭼq��n9���2�n���u�+���p����2�k�6���1����(�p�)���������P����*����9�1���̍��9�n�=1����a���̽n���P�q��n9���2�pn��
]��.��<(Note:���when��p�j�N�@��,�L���(�p�)��is�c��ffonsider�e�d�G�to�b��ffe����0�35�and��U���̽p�����is�use��ffd�inste�ad�of��T���̽p�����which�is�c�al���le�d�A��2tkin-L�ehner�op�er�ator.)������(2)�35if��(�n;���m�)�UR=�1�,�35then��T���̽nm���f�=�UR�T���̽n���P�T���̽m����.������(3)�35if��p�UR�6�j�N�@��,�35then��T��O��p�����l�����=�UR�T��O��p�����l��1���"�T���̽p��r�����p����2�k�6���1��U�<�p�>�T��O��p�����l��2����.������(4)�35if��p�j�N�@��,�then��T��O��p�����l�����=�UR(�T���̽p���]�)����2�l��!��.��6����The�.last�form��rula�in�the�theorem�can�b�S�e�pro�v�ed�b�y�comparing�the�co�S�ecien�ts�of��q��n9���2�p���-:�l��1����;�in���b�S�oth��sides�of�the�follo��rwing�formal�iden�tit�y:��!U]����h� ��f\� ��������t�.�1��
���q�N�����X���ҁ��p�̽n�=1����ږ�T���̽n���P�q��n9����n������f\�!�����ǿj�T���̽p����=���������1��
���ԟ����X���ҁ��UR�n�=1���Z�T���̽pn��	ﭼq��n9����n���1�+����p�����k�6���1��U�<�URp�>���������1��
���ԟ����X���ҁ���n�=1����T���̽n���P�q��n9����pn��
]�:��"'Ս�F��Vor�)�example,�9Pthe�co�S�ecien��rt�of��q��n9���2�p��	_+�in�the�LHS�)�is��T���̽p���]�T���̽p���.���On�)�the�other�hand,�the�co�S�ecien��rt�of����q��n9���2�p��	 >�in��the�RHS�is��T��ؽp�����2������+����p����2�k�6���1��U�<�URp�>�T���̺1����.�8�Th��rus,��w�e�ha�v�e������!L�T��ؽp�����2���GV�=�UR(�T���̽p���]�)�����2��j������p�����k�6���1��U�<�p�>�I��d����where�꨼<�URp�>��should�b�S�e�considered�to�b�e�a�n��rull�map�if��p�j�N�@�.�����3.�32The�ffOld�F���forms�������Supp�S�ose�J��M�@�j�N��.��Let��f��Q�2�UR�S���̽k��#��(���̺1����(�M��)).��Then�for��d��suc��rh�that��d�j�����Fu��艽N��33�콉fe	��'�M�����jV�,�j��f�G��(�d�W�)�UR�2��S���̽k��#��(���̺1����(�N�@�)).��Th��rus���w��re��ha�v�e�a�map����������̽M���B�:����'G�����M���	���UR�d�j���������Ȉ�N��33�x|�fe�ܟ���M���������S���̽k��#��(���̺1����(�M�@�))�UR�!��S���̽k���(���̺1����(�N�@�))�:��"����The�~old�part�of��S���̽k��#��(���̺1����(�N�@�))�is�dened�as�the�subspace�generated�b��ry�the�images�of�����̽M���n�for����M�@�j�N��,�꨼M��6�6�=�UR�N��.������Example��3���DUQ����̽M��)�is��9not�inje��ffctive.�	�tConsider�the�c�ase�that��k�J&�=��	12�,��y�M�#�=��p��9�and��N��=��	�p����2�2����.����S���̽k��#��(���̺1����(�p�))�35�c��ffontains��(��W�)��and��(�p��)�.�fiBut�����̽p�����maps�b��ffoth�of�them�to��(�p��)��in��S���̽k��#��(���̺1����(�p����2�2���))�.��������21�������K����b�{�K������{����Theorem��9���E���Supp��ffose�Ko�p�\?�)���
msbm10�-��N�@��.��Consider��f���;���g��x�2��S���̽k��#��(���̺1����(�N�@�))�.��Then��f��n�and��g�n9�(�p�W�)��ar��ffe�b�oth�in�����S���̽k��#��(���̺1����(�N�@�p�))�.�fiThen�35we�have���ፒ���f�G��j�U���̽p����=�UR(�f��j�T���̽p���]�)������p�����k�6���1����(�p�)(�f�G��(�p�W�))����and�����Wj�g�n9�(�p�W�)�j�U���̽p����=�UR�g��(���)���<��wher��ffe�35�f�G��j�T���̽p�����is�c�onsider�e�d�in��S���̽k��#��(���̺1����(�N�@�))�.���K��Pro�of.�8�Let�꨼f��Q�=��UR�����P����*���
��1���̍�
㍽n�=1��� hW�a���̽n���P�q��n9���2�n����.�Then,�considering�in��S���̽k��#��(���̺1����(�N�@�)),�w��re�ha�v�e�� ګ������f�G��j�T���̽p����=���������1��
���ԟ����X���ҁ��UR�n�=1���Z�a���̽pn��	ﭼq��n9����n���1�+����p�����k�6���1����(�p�)���������1��
���ɀ�����X���ҁ�����n�=1���ȼa���̽n���P�q��n9����pn��
]�:��";b��Also,��considering�in��S���̽k��#��(���̺1����(�N�@�p�)),�w��re�ha�v�e�����Ѽf�G��j�U���̽p����=���������1��
���ԟ����X���ҁ��UR�n�=1���Z�a���̽pn��	ﭼq��n9����n����:��!����Th��rus,��w�e�ha�v�e�����#<�f�G��j�U���̽p����=�UR(�f��j�T���̽p���]�)������p�����k�6���1����(�p�)(�f�G��(�p�W�))�:���<��No��rw,��let��g�Ë�=��UR�����P����*���
��1���̍�
㍽n�=1��� hW�b���̽n���P�q��n9���2�n����.�8�Then��!]���:�g�n9�(�p�W�)�j�U���̽p����=���UR��f\� ���������1��
���	������X���ҁ��?��n�=1���Dȼb���߽n=p��/��q������n������f\�!���E�C�j�U���̽p���=�UR�g��(��W�)��";b�where�꨼b���߽n=p����=�UR0�unless��p�j�n�.��'���F��aGebruary�z�7,�1996��b#��Scrib�S�e:�8�Amo�d��Agashe,��<[email protected]>��)�����W��Ve�{Sare�in�the�pro�S�cess�of�sho��rwing�that�the�Hec�k�e�op�S�erators��T���̽p��B��acting�on�the�space�of�cusp���forms�꨼S���̽k��#��(���̺1����(�N�@�))�are�not�necessarily�semisimple�if��p�UR�j��N��.������Recall�� from�last�time�that�if��M�e��j�%�N�<�then�for�ev��rery�divisor��d��of��M���=��X�N�@�,�?>w�e�had�a�map����S���̽k��#��(���̺1����(�M�@�))�UR�!��S���̽k���(���̺1����(�N�@�))��giv��ren�b�y��f�G��(��W�)�UR�7!��f��(�d�W�).������Note�v�that�the�v��X�arious��f�G��(�d�W�)'s�are�linearly�indep�S�enden��rt�o�v�er��C�,���b�S�ecause�the�F��Vourier�ex-���pansion��of��f�G��(�d�W�)�starts�with��q��n9���2�d��M�.������Let��ؼf�<׹b�S�e�an�eigenfun��rtion�for�all�the�Hec�k�e�op�S�erators��T���̽n��	�(�in��S���̽k��#��(���̺1����(�M�@�)).�WpLet��p��b�e�a�prime���not�I�dividing��M�@�.�	VSo��f���j���T���̽p��
r>�=��af����where�I��a��=��a���̽p���]�(�f�G��)�and��f���j�<�p�>�=���(�p�)�f����where���(�p�)�is���the�ٛc��rharacter�asso�S�ciated�to�the�mo�dular�form��f�G��.��Note�that�one�can�pro��rv�e�ٛthat�if��f�!��is�an���eigenfunction�J�for�the��T���̽n���P�'s�then�it�is�an�eigenfunction�for�the�diamond�op�S�erators�also�(or���alternativ��rely��V,���mak�e�m	it�part�of�the�denition�of�eigenform).��Let��N�t"�=�3>�p����2������M���with���FͿ��1.��W��Ve�������22����#��K����b�{�K������{��will�ilo�S�ok�at�the�action�of�the��p����2�th���$�Hec��rk�e�iop�erator��U���̽p���ƹin��S���̽k��#��(���̺1����(�N�@�))�on�the�images�of��f�Vh�under����the��maps�describ�S�ed�ab�o��rv�e.�8�Let�꨼f���̽i��dڹ(� �W �)�UR=��f�G��(�p����2�i��� �W �)��for�0�UR���i���� ���.�8�As��w��re�sho�w�ed�earlier,������ r�f��Q�j�UR�T���̽p�� ��=���������X������a���̽np�� ﭼq��n9�� ��n���1�+�����(�p�)�p��� ��k�6���1����� �����X����"qӼa���̽n���P�q��n9�� ��pn�� ]�:����So��������af��Q�=�UR�f���̺0��V�j��U���̽p��r�+�����(�p�)�p��� ��k�6���1���f���̺1����:����Th��rus,�������f���̺0��V�j�UR�U���̽p�� ��=��af���̺0��j�������(�p�)�p��� ��k�6���1���f���̺1����:����F��Vrom�)�last�time,�P$w��re�ha�v�e��f���̺1��V�j�UR�U���̽p����=��f���̺0����:�)��In�fact,�P$in�general,�one�can�see�easily�that��f���̽i���,�j�UR�U���̽p�� ��=��f���̽i��1�����for�꨼i�UR���1.������So����U���̽p��i�preserv��res�the�2-dimensional�space�spanned�b�y��f���̺0��a��and��f���̺1����.� �The�matrix�of��U���̽p��i�(acting���on��the�righ��rt)�with�this�basis�is�giv�en�(from�the�equations�ab�S�o�v�e)�b�y:��% ���������f\� �����d�����&�a���r�1��������o:���(�p�)�p����2�k�6���1����r�0�����R��f\�!����"���The��c��rharacteristic�p�S�olynomial�of�this�matrix�is��x����2�2��j������ax��+��p����2�k�6���1����(�p�).������There��cis�the�follo��rwing�striking�coincidence:�4=Let��E��z�b�S�e�the�n�um�b�S�er� eld�generated�o�v�er��Q����b��ry��Hthe�co�S�ecien�ts�of�the�F��Vourier�series�expansion�of��f��G�and�let����b�S�e�a�prime�ideal�of��O���̽E�� �йlying���o��rv�er��some�rational�prime��l�C��.�8�Then�w��re�ha�v�e�a�Galois�represen�tation�� ����ES����̽��ʬ�:�UR�Gal�C��(�����fe #�� n��Q��� #��=�Q�)��!���GL������̺2����(�E���̽��uZ�)����If� {�p�UR�6�j�N�@�l�O �then�����̽���չis�unrami ed�and�also��det����̽��uZ�(�F���r�S�ob���̽p���]�)�=���(�p�)�p����2�k�6���1�� ��and��tr�_ ���̽��uZ�(�F���r�S�ob���̽p���]�)�=��a���̽p���(�f�G��)�=����a�.��0Th��rus�*mthe�c�haracteristic�p�S�olynomial�of�����̽��uZ�(�F���r�ob���̽p���]�)�is��x����2�2�������ax��+��p����2�k�6���1����(�p�),�:_the�*msame�as�that���of��the�matrix�of��U���̽p���]�!������A��|question���one�can�ask�is:���Is��U���̽p�� n �semisimple�on�the�space�spanned�b��ry��f���̺0�� f��and��f���̺1����?�l�The���answ��rer��is�y�es�if�the�eigen�v��X�alues�of��U���̽p����are�di eren�t.������No��rw,��Lthe���eigen�v��X�alues�are�the�same�i �the�discriminan�t�of�the�c�haracteristic�p�S�olynomial���is���zero�i.e.�N��a����2�2�� D<�=��84��(�p�)�p����2�k�6���1�����i.e.��a��=�2�p����#B��33�k� ���1��33�� �fe �ڟ�����2����[email protected]�����where������is�some�square�ro�S�ot�of���(�p�).�N�Here�is�a��i�curious�fact:�c�the�Raman��rujan-P�etersson�conjecture�pro��rv�ed�b�y�Deligne�sa�ys��j�a�j�y���2�p����#B��33�k� ���1��33�� �fe �ڟ�����2����[email protected]�;� �th��rus���the�rlab�S�o��rv�e�equalit�y�is�allo�w�ed�b�y�it,��xso�w�e�do�not�get�an�y�conclusion�ab�S�out�the�semisimplicit�y���of�꨼U���̽p���]�.������Let�VAus�no��rw�sp�S�ecialize�to��k��o�=�UR2.�hW��Veil�has�sho�wn�that�����̽��uZ�(�F���r�S�ob���̽p���]�)�is�semisimple.�hTh�us�if�the���eigen��rv��X�alues��\of��U���̽p�����are�equal,��Ithen�����̽��uZ�(�F���r�S�ob���̽p���]�)�is�a�scalar.�g�Edixho�v�en�pro�v�ed�that�it�is�not.�g�So���the�eigen��rv��X�alues�of��U���̽p���e�are�di eren�t�and�hence��U���̽p���e�is�semisimple�in�this�case.��So�this�example���(for��k=2)�do�S�es�not�giv��re�us�an�example�of��U���̽p����b�eing�not�semisimple.������There�i is�the�follo��rwing�example�giv�en�b�y�Shim�ura�whic�h�sho�ws�that�the�Hec�k�e�op�S�erator����U���̽p�� ���need��bnot�b�S�e�semisimple.�� Let��W�^(�denote�the�space�spanned�b��ry��f���̺0����;���f���̺1�� |f�and��V�Xҹdenote�the���space��spanned�b��ry��f���̺0����;���f���̺1���;�f���̺2���;�f���̺3���.�R:�U���̽p���x�preserv��res��b�S�oth�spaces��W���and��V��p�,��8so�it�acts�on��V�N8=W��ƹ.�The��� ����23����01��K����b�{�K������{��action��3is�giv��ren�b�y����w}�fe ��� ����f���̺2�����1�7!��UR��w}�fe ��� ����f���̺1����5P�=�UR0�and����w}�fe ��� ����f���̺3�����7!��UR��w}�fe ��� ����f���̺2�����where�the�bar�denotes�the�image�in��V�N8=W��ƹ.�(�Th��rus����the��matrix�of��U���̽p����on�the�space��V�N8=W��n�is��"Y���������f\� �����d������0�����1���������0�����0����������f\�!����!S㍹whic��rh�� is�nilp�S�oten�t,�|and�in�particular�not�semisimple.��CIf��U���̽p�� �{�w�ere�semisimple�on��V�p��then�it���w��rould�{�b�S�e�semisimple�on��V�N8=W� ��also;��Lbut�w�e�ha�v�e�just�sho�wn�that�it�is�not.��(Th�us��U���̽p�� C �is�not���semisimple��on��V��p�,�and�hence�not�on��S���̺2����(���̺1���(�M�@�))��(b�S�ecause��V���is�in��rv��X�arian�t��under��U���̽p���]�).��'����W��Ve� next�discuss�the�structure�of�the��C�-algebra��T�5��=��T�����C�� ���generated� b��ry�the�Hec�k�e�and���diamond��op�S�erators�and�the�structure�of��S���̽k��#��(���̺1����(�N�@�))�as�a��T�-mo�dule.������First��w��re�consider�the�case�of�lev�el�1�i.e.� �'�N��[�=��w1.�Then�����̺1����(1)�=��S���L���̺2���(�Z�).� �'All��the����T���̽n���P�'s���are�diagonalizable.����S���̽k�� zG�=�V��S���̽k��#��(���̺1����(�N�@�))�has�a�basis�of��f���̺1���;���::::;�f���̽d�� a��of���normalized�eigenforms���where�L�d��]�=��dim�(�S���̽k��#��).� _�Th��rus��S���̽k������P��� �����԰��� �׹=�����٣�C����2�d�� ,��as�a��C�-v�ector�space.� _�Then�w�e�ha�v�e�the��C�-algebra���homomorphism�E��T���!��C����2�d�� %O�giv��ren�b�y��T�E��7!���(����̺1����;���::::;����̽d��ߨ�)�where��f���̽i�� Ϳj���T�E��=�����̽i��dڼf���̽i���.� I�It�is�injectiv��re���b�S�ecause��uif�the�image�of��T��;�is�zero,���then�it�kills�all��f���̽i��]O�i.e.�bFall�of��S���̽k�� �i.e.�it�is�the�zero�op�S�erator.���The�uUmap�is�surjectiv��re�b�S�ecause��T��has�dimension��d�.���Th�us�as�a��C�-algb�S�ebra,���T�����P���A\����԰���ZD�=�������C����2�d��ߨ�.���Next,���w��re�_claim�that�the�mo�S�dular�form��v��=��~�f���̺1��}��+����:::��+��f���̽d����generates�_�S���̽k�� )�as�a��T�-mo�dule.��This�follo��rws���b�S�ecause�L�under�the�map��S���̽k������P���x����԰����̹=�����#��C����2�d��ߨ�;���v�Ë�7!�UR�(1�;�::::;��1)�L�and�our�statemen��rt�is�just�the�trivial�fact�that���(1�;���::::;��1)��generates��C����2�d���P�as�a��C����2�d��ߨ�-mo�S�dule�(acting�comp�onen��rt-wise).������Th��rus��i�S���̽k�� ���is�free�of�rank�1�as�a��T�-mo�S�dule.��$W��Ve�already�kno�w�that��S���̽k������P���	�"����԰���

�=�����	�H��Vom�(�T�;����C�)�as����T�-mo�S�dules.���Th��rus�6<�T�����P���UR����԰���n:�=��������H��Vom�(�T�;����C�)�as��T�-mo�dules.���In�fact�the�isomorphism�is�canonical�since���the�꨼f���̽i��dڹ's�are�normalized.�8�W��Ve�remark�that��v�X�in�fact�lies�in��S���̽k��#��(�Q�).������Next,��w��re�deal�with�the�general�case�where�the�lev�el�is�not�necessarily�1.������First���w��re�need�to�talk�ab�S�out�newforms.��ERecall�the�maps��S���̽k��#��(���̺1����(�M�@�))�UR�!��S���̽k���(���̺1����(�N�@�))���for�ev��rery���divisor�мd��of��M���=��X�N�^��men��rtioned�at�the�b�S�eginning�of�this�lecture.��XThe�old�part�of��S���̽k��#��(���̺1����(�N�@�))�is���dened���as�the�space�generated�b��ry�all�the�images�of��S���̽k��#��(���̺1����(�M�@�))�for�all��M���j�k/�N���;���M��6�=��N�8d�under���these���maps.���The�new�part�of��S���̽k��#��(���̺1����(�N�@�))�can�b�S�e�dened�in�t��rw�o���dieren�t�w�a�ys.���Firstly�w�e���can�ߖdene�it�as�the�orthogonal�complemen��rt�of�the�old�part�with�resp�S�ect�to�the�P�etersson���inner��pro�S�duct.�J{There�is�also�an�algbraic�denition�as�follo��rws.�There�are�certain�maps�going���the��hother�w��ra�y:��S���̽k��#��(���̺1����(�N�@�))��L�!��S���̽k���(���̺1����(�M�@�))��hfor��M�)0�j��L�N���;���M��6�=��N�@�.�� The��hnew�part�is�the�space���killed�$Nunder�all�these�maps.���The�space�of�newforms,�2�denoted��S���̽k��#��(���̺1����(�N�@�))���̽new����is�lik��re��S���̽k���((1))���in��Othe�sense�that�all�the��T���̽n���P�'s�(including��U���̽p���]�)�are�semisimple�and�there�is�a�basis�consisting�of���newforms.�8�A��form�of�lev��rel��N�+��is�said�to�b�S�e�new�of�lev�el��N�+��if�it�is�in��S���̽k��#��(���̺1����(�N�@�))���̽new���]�.������Next,���one�#dcan�sho��rw�that�the�map�������L���?؟��M��"�j�N�a>;M���N��?ֺ�S���̽k��#��(���̺1����(�M�@�))���̽new����!� �S���̽k���(���̺1���(�N�@�))�#dgiv��ren�b�y�����f�G��(� �W �)�; �7!��f��(�d �W �)�q�for��d�; �j�����Fu��#��N��n>�콉fe ��'�M������is�injectiv��re�(See�W.-C.�W.Li,��\Newforms�and�functional�equations,���Math.�(NAnnalen,���212(1975),�285-315).�Note���that�an�eigenform�in�one�of�the�subspaces�of�the���source�I�need�not�b�S�e�an�eigenfun��rtion�for�all�the�op�erators�in�the�image.�>If��f����is�a�newform,�i�then���let��O�M���̽f�� Pn�denote�its�lev��rel�(i.e.�3�f�!N�is�new�of�lev�el��M���̽f��w �).�3Let��S��&�b�S�e�the�set�of�newforms�of�w�eigh�t��k����and��some�lev��rel�dividing��N�@�.�8�Let�����8(�v�Ë�=����<F�����X��� 8獑UR�f����2�S�����f�G��(������ō�H��N��33�Qm�fe�%�  �M���̽f������;�� �W �)�:��� �����24����@��K����b�{�K������{��Then�~�one�can�sho��rw�that��S���̽k��#��(���̺1����(�N�@�))�is�free�of�rank�1�o�v�er��T�����C�� �with��v��8�as�the�basis�elemen�t.����Also��one�can�sho��rw�that��v�X�has�rational�co�S�ecien�ts.��(V��F��aGebruary�z�9,�1996��b#��Scrib�S�e:�8�J�� anos��Csirik,��<[email protected]>��-�Ս�Final��commen��ts�ab�out�Hec�k�e�algebras��@��Recall�^kthat�for�the�case��\=��S���L�(2�;����Z�),�{\if�^kw��re�set��f���̺1����;��:�:�:��ʜ;�f���̽d�� >�to�b�S�e�the�normalized�eigenforms���(newforms�XEof�lev��rel�1),���then�they�ha�v�e�p�S�ossibly�complex�co�ecien��rts�but�in�an�y�case��f�f���̽i��dڿg����is�H� nite�and�stable�under�automorphisms�of��C��and�all�the�co�S�ecien��rts��a���̽n���P�(�f���̽i��dڹ)�lie�in�some���n��rum�b�S�er�%u eld.��FF��Vurthermore�this� eld�is�totally�real:��yto�sho��rw�this�w�e�used�that�since�the�set���is��stable�under�conjugation,��2it�suces�to�sho��rw�that�all�the��a���̽n���P�(�f���̽i��dڹ)�are�real,�whic��rh�follo�w�ed�b�y���remarking�$�that�they�are�eigen��rv��X�alues�of�the�op�S�erators��T���̽n��	��whic�h�are�self-adjoin�t�with�resp�S�ect���to��the�P��retersson�inner�pro�S�duct.������More�(�generally��V,�8Dlet��f���2���S��׹(���̺1����(�N�@�))�b�S�e�a�normalized�eigenform�of�c��rharacter��"�.��"Then��a���̽n��	gP�=����"�(�n�)����&�fe�Q�]ڍ�a���̽n�����Q�.������Note�i�that�the�algebra��T�����Q��
F\�generated�b��ry�the��T���̽i�����o�v�er��Q��con�tains�the�diamond�brac�k�et���op�S�erators:�s�the��form��rula�relating��T��ؽp�����2���z�and�(�T���̽p���]�)����2�2��
H�tells�us�that�the�dierence�is��p����2�k�6���1���"�(�p�),��pso����"�(�p�)�W^�2��T�����Q���v�.�
�Using��Diric��rhlet's�Theorem�on�primes�in�arithmetic�progressions,� )for�an�y��d����relativ��rely��prime�to��N�+��w�e�can�nd�a�prime��p�UR���d���(�mo�S�d���B�N�@�),��so��"�(�d�)�UR=��"�(�p�)��2��T�����Q���v�.������If�.Wthe�space�of�mo�S�dular�forms�has�dimension�1,�Cthen�it�is�spanned�b��ry�a�(normalized)���eigenform��with�rational�co�S�ecien��rts,�so�the�eigen�v��X�alues�are�all�in��Q�.������The���next�simplest�example�is��k��o�=�UR24,�]whic��rh�is�the�smallest�w�eigh�t�suc�h�that�the�dimension���is���more�than�one.�أThere�are�t��rw�o���eigenforms,��whic�h�are�conjugate�to�eac�h�other.�أIf��f��Q�=��UR�����P��㋼a���̽n���P�q��n9���2�n�����is��;an�eigenform,�� then��Q�(��:���:�:��ʞa���̽n����N�:���:�:��r�)�[h=��Q�(������p���
����z�#?�	�l��144169����-?�).�C�In��;fact��S���̺24���C�is�spanned�b��ry�����2�2���?�and�����2�2����j�T���̺2���.���(Note��5that�����2�2��	r9�is�denitely�not�an�eigenform�since�its��q�n9�-co�S�ecien��rt�is�0.)���The�action�of��T���̺2�����on�'�S���̺24��'��with�resp�S�ect�to�this�basis�is�describ�ed�b��ry�a�t�w�o�b�y�t�w�o�matrix�of�trace�1080�and���determinan��rt��#��2����2�10��	�3����2�2����2221.��RF��Vor�high��k�g�,��Beigenforms�tend�to�form�a�single�orbit;�)aho�w�ev�er,��Bno���pro�S�of��is�kno��rwn�for�this.������F��Vor�mev��rery�newform��f�G��,�^let��E���̽f��	���b�S�e�the�n�um�b�S�er�eld�generated�b�y�its�co�S�ecien�ts.��/Let��b�S�e���a��set�of�represen��rtativ�es��for��f�G��'s�mo�S�dulo��Gal�C��(����!�����Q���
#��=�Q�).�8�Dene�����������T�����Q�������!�!������!�E���̽f���������Ҝ9�T�����!�7!������!����̽T����;������with�꨼f�G��j�T���=�UR����̽T����f��.�8�Th��rus��T���̽n�����7!�UR�a���̽n��	���so�this�map�is�surjectiv�e.�������In��fact�the�induced�����s��T�����Q��1ȿ!����Tu�����Y���8獑UR�f����2������E���̽f���������25����VO��K����b�{�K������{��is�6�an�isomorphism�of��Q�-algebras:��	it�is�injectiv��re�since�if��T�؂�dies�on�the�image�then�it�acts�as����zero��on��f��Q�2�UR��and�so�on�all�of�f,��b��ry�the�rationalit�y�of�Hec�k�e�op�S�erators:��5g���2����h�(�g�n9�j�T��ƹ)�UR=�����2����S�g��j�T��ƹ.�6�(And���if��an�op�S�erator�acts�as�zero�on�ev��rerything,�then�it�is�zero.)������Here��w��re�used�the�fact�that�there�w�ere�no�oldforms�around.������F��Vor� �example,�. consider��S���̺2����((�N�@�))�for��N�a��prime.���Then��S���̺2���((1))�is�empt��ry��V,�. hence�w�e�get�an���isomorphism�����3\�T�����Q������P���1ȿ���԰���J��=������q�E���̺1��j�������:���:�:�������E���̽t���ʼ;���~��with�[the�righ��rt�hand�side�a�pro�S�duct�of�totally�real�n�um�b�S�er�elds.��E�t��>��1�[is�p�ossible,�w=e.g.��Efor����N��6�=�UR37,���T�����Q��1ȹ=��Q������Q�.������In��general,�oldforms�complicate�the�situation.��&���Final��commen��ts�ab�out�Hec�k�e�algebras��@��W��Ve'll��only�treat�the�case��k��o�=�UR2.�8�Then�(for��a�congruence�subgroup),���&���pؼS���̺2����()�UR=��H���V����0���Z�(�X���̺����;����
�����1���)���where���X���̺��	$F�=�UR���n��H��[����n��P����2�1����(�Q�),�� with���n��P����2�1����(�Q�)�b�S�eing�the�set�of�cusps�that�w��re�need�to�adjoin����to��mak��re�it�a�compact�Riemann�surface.������Therefore��������dim��·7�S���̺2����()�UR=��g�n9�(�X���̺����)�:�������Example.�kҼS���L�(2�;����Z�)��'�n��P����2�1����(�Q�)�N�has�just�one�p�S�oin��rt.� e_The�pro�of�in��rv�olv�es�N�the�Euclidean�� �algorithm:��an��ry��4elemen�t�of��P����2�1����(�Q�)�can�b�S�e�written�as�����f\� �����d�����x������� y����� t2��f\�!���*;�with��x��and��y�Jm�relativ�ely�prime��U^�in��rtegers.��By�2�the�Euclidean�Algorithm,�E w�e�can� nd��a��and��b��in�tegers�suc�h�that��ax���+��by�>��=��c1.���Then�� ������ ��f\� �����d����qg�a���� 6b���������̿�y����Fx�������T��f\�!����������f\ �����d����Ȫ�x�������� �y������u���f\�!������=���UR��f\� �����d���?��1�������?�0����� ���f\�!���� �����T��Vo��calculate��g�n9�(�X���̺����),� �use�the�follo��rwing�co�v�ering�(recall�that��X���̺�� ғ�is�the�compacti cation�of�������n��H�꨹w��re�obtain�b�y�adjoining�the�cusps)���&�����X���̺��$F�!�UR�X���ߺ(1)���p�;����k��reeping��in�mind�the�isomorphism�������i��j�%�:�UR�X���ߺ(1)�������M�!������M�P�����1����(�C�)������������(�i;���;��1�)������M�7!������M�(1728�;����0�;��1�)�����The��only�ramication�in�our�co��rv�ering��o�S�ccurs�ab�o��rv�e��the�p�oin��rts�0�;����1728�;��1�.��������Example.�
+j�Let��H�£=����̺0����(�N�@�).���The�degree�of�the�co��rv�ering�is�(�P���S���L�(2�;����Z�)�£:����̺0����(�N�@�)�=�<ȿ��1)���whic��rh��is�the�n�um�b�S�er�of�cyclic�subgroups�of�order��N���in��S���L�(2�;����Z�=��X�N�@��Z�).�;?W��Ve�ha�v�e�a�co�v�ering����Y���̺0����(�N�@�)��!��Y���ߺ(1)���b�where�N�Y���̺0���(�N�@�)�parametrizes�elliptic�curv��res��E�	�with�a�cyclic�subgroup�of�order����N�@�,�s��C�윿���E���[�N��]�����P�������԰���(�=�����uS�Z�=��X�N��Z��S���Z�=n�Z�XT�up�to�isomorphism;��*and��Y���ߺ(1)��Ĺparametrises�elliptic�curv��res.�������26����d���K����b�{�K������{��The�	"isomorphism�(�E��;���C���̺1����)�����P����3����԰�����=�����g�(�E�;�C���̺2����)�	"is�an�automorphism�����of��E��9�with����¹:��3�C���̺1��I7�7!��C���̺2���.��OUsually������h�=�UR��1,��unless��j�%�=�1728�;����0�;��1�.������If�]w��re�understand�ramication,�
Jw�e�can�use�the��R��2iemann-Hurwitz�L�formula�.���The�follo�wing���mnemonic�Ǌw��ra�y�of�thinking�ab�S�out�it�is�due�to�N.�Katz.�-+The�Euler�c�haracteristic�(alternating���sum��of�the�dimensions�of�cohomology�groups)�should�b�S�e�though��rt�of�as�totally�additiv�e:�������ͼ�(�A�����������a������B���)�UR=���(�A�)���+���(�B��)�:������If���X��e�is�a�Riemann�surface�of�gen��rus��G�,�X0��(�X��)���=�2��1���2�G�.��A��bsingle���p�S�oin�t�has�Euler����c��rharacteristic�꨼�(�P��ƹ)�UR=�1.�8�Hence�����u���(�X��+�n���f�P���̺1����;����:�:�:��ʜ;���P���̽n���P�g�)�UR=�2����2�G����n:����Therefore�����JѼ�(�X���ߺ(1)��V�n���f�0�;����1728�;��1g�)�UR=���1���and����t߼�(�X���̺��y��n���f��p�S�oin��rts��o�v�er�1728�;����0�;��1��pYVg�)�UR=�2����2�g����n����if�e��n��p�S�oin��rts�lie�o�v�er��0�;����1728�;��1��8L��.��zIf�the�co�v�ering�map��X���̺��	�i�!�&u�X���ߺ(1)����has�degree��d�,��>then�w�e�can���think��of�the�top�space�as��d��copies�of�the�b�S�ottom�space,�so������V��d������(��1)�UR=�2������2�g����n����and��therefore������2�g������2�UR=��d����n��=��d����n���̺0��j����n���̺1728��*����n�����1��	�:�������Example.���Let�;���~=�(�N�@�).�,hWhat�happ�S�ens�o��rv�er��j��Q�=��~0?�,hThis�corresp�S�onds�to�an�elliptic���curv��re�꨼E����with�an�automorphism����7�of�order�three.�8�Let��N��6>�UR�3.������F��Vor�*�an��ry�(�E��;���P�S�;�Q�),�Q)w�e�*�ha�v�e�(�E��;�����P�S�;��Q�)�*�and�(�E��;��������2�2��ӓ�P�S�;������2�2��ӓ�Q�)�*�whic��rh�are�isomorphic�to�it�and���hence�()they�are�the�same�p�S�oin��rt�on��X���̺����.��dSo��E��@�only�has�one-third�the�usual�n�um�b�S�er�of�p�oin��rts���lying���o��rv�er�it,����n���̺0��#k�=�cg�d=�3.�Q�(Except�if�the�ab�S�o�v�e�three�p�S�oin�ts�are�equal,���i.e.,���}�xes��E���[�N�@�].�Q�This���can't��happ�S�en�since��N��6>�UR�3.)������Similarly��w��re�get��n���̺1728���b�=�UR�d=�2.������T��Vo�1Vdetermine�the�degree��d�,��x��E��m�and�coun��rt�the�p�S�oin�ts�lying�o�v�er�it:��=these�are�all�of���the�]Xform�(�C�=�Z��
����W�Z�;����1�=n;��=��X�N�@�)�]Xwith�W��Veil�pairing��e����2�2��I{i=��N�����and�all�suc��rh�o�S�ccur,��so�w�e�need���to���coun��rt�the�n�um�b�S�er�of��P�;���Q��;�2��E���[�N�@�]���whic��rh�form�a�basis�of��E��[�N�@�]�and��e���̽N��D�h�P�S�;���Q�i��;�=��e����2�2��I{i=��N��0L�.���This���giv��res�us�the�order�of��S���L�(2�;����Z�=n�Z�).�
�Ho�w�ev�er,��r(since��N�WH�6�=�d2)�w�e�ha�v�e�to�tak�e�in�to���accoun��rt�that�(�E��;���P�S�;�Q�)�����P���������԰���ǂ�=�������(�E�;���P�S�;���Q�)�but�they�are�not�equal�so�the�degree�of�the�co��rv�ering���is��#(�S���L�(2�;����Z�=��X�N�@��Z�))�=�2.������W��Ve���also�ha��rv�e���d�UR�=�(�P���S���L�(2�;����Z�)�:�(�N�@�)�=�((�N��)�b�\��1))�since�(�N�@�)��n��S���L�(2�;����Z�)�����P���UR����԰���n:�=��������S�L�(2�;��Z�=��X�N�@��Z�).������So��w��re�ha�v�e�established�that����}��2�g������2�UR=��d����n���̺0��j����n���̺1728��*����n�����1��UZ�=��d=�6����n�����1��	�:��������27����r4��K����b�{�K���������T��Vo�IYdetermine��n�����1��	�,�anote�that��S���L�(2�;����Z�)�acts�on��P����2�1����(�Q�)����3�1��=�����f\� �����d����,�1��������,0������(��f\�!���#�Թ.�T�The�IYstabilizer�(�N�@�)�����1�����of���n&��f\� �����d���Xҹ1�������X�0�����8Ο�f\�!���&���is�n&�U��6�=�UR��������f\� �����d���ꪹ1���ʦ��������ꪹ0���ک1�����)ʧ��f\�!���5#y�So�the�index�((�SL���
ދ��̺2�����(�Z�)�=��X���1)�����1��UZ�:�(�N�@�)�����1��	�)�=��N��
�is�n&the�ramication��
�degree��of�a�p�S�oin��rt�o�v�er��1�,�so��n�����1��UZ�=�UR�N�F:=d�.�������Hence��2�g�n9�(�X��(�N�@�))������2�UR=��d=�6������d=��X�N��.��'cT��F��aGebruary�z�12,�1996��b#��Scrib�S�e:�8�J�� anos��Csirik,��<[email protected]>���*����Plug:�|�A��useful��reference�for�the�next�lecture�is�Andrew�Ogg:��R��ffational�Rxp�oints�on�c�ertain���el���liptic�35mo��ffdular�curves�꨹(1972).������F��Vrom��last�lecture's�results�on�easily�deduces�that�������g�n9�(�X��(�N�@�))�UR=�1���+������ō�!�d���۟Qm�fei����12�N�����z��(�N�댿��6)�:��ϖ�����Example.�ȉ�Let���N�縹=���5�;����7.�If��N�[o�is�prime�then�the�degree�of�the�co��rv�ering�is�����fh��M��(�N���"��-:�2�����1)(�N���"��-:�2����N��"�)��M���ʉfe=����'�����N��"��1�����DS�.���Therefore����N��-�=�nI5�giv��res��d��=�60�and��g�܂�=�0�(the�Galois�group�of�the�co��rv�ering���in�this�case�is����A���̺5����.)��YSimilarly��V,����N��=��17��{yields��g�;j�=�3�and��d��=�168.��Y(Remark:��F��Vor��p����5�prime,���the�group����S���L�(2�;����Z�=p�Z�)�=������1�UR=��L���̺2����(�p�)��is�simple.)�������Example.�8�What��is�the�gen��rus�of��X���̺0����(�N�@�),�for��N�+��a�prime?��	����It��is�an�exercise�to�sho��rw�that�there�are�t�w�o�cusps:������f\� �����d���
a�1�������
a0������]��f\�!���+ȭ�=��1��and�����f\� �����d���Ⱦ�0�������Ⱦ1���������f\�!���)�
�=�0.�The���co��rv�ering�bu�X���̺0����(�N�@�)�UR�!��X��(1)�is�easily�seen�to�ha��rv�e�degree��N��[�+��w1,�}�since�an�elliptic�curv�e�has��N��[�+��w1���cyclic��subgroups�of�order��N�@�.�8�Here��1��is�unramied�and�0�has�ramication�index��N��.�8�Hence������k2�g������2�UR=��N�댹+�1����n�����1�������n���̺1728��*����n���̺0�����with�꨼n�����1��UZ�=�UR2,��n���̺1728��j��appro��rximately��d=�2�and��n���̺0�����appro�ximately��d=�3.�������T��Vo��fcalculate��n���̺0����,���w��re�need�the�n�um�b�S�er�of�isomorphism�classes�(�E��;���C�ܞ�)�with��E�l}�xed�with����E��nd�(�E��)�$�=��Z�[����̺6����].������̺6��$��acts�d�on�the�set�of��C�ܞ�'s.�Since���1�is�acting�trivially��V,��)w��re�really�ha�v�e�an���action��of�����̺3�����with�(�E��;���C�ܞ�)�����P���UR����԰���n:�=�������(�E�;���C�ܞ�)�����P���UR����԰���n:�=�������(�E�;�������2�2����C�ܞ�)��with���ҩ�some�third�ro�S�ot�of�unit��ry��V.������If���w��re�consider��E�	i�=�UR�C�=�O�E�with��O����=��Z�[(��1�Fv+��i������p���
����z����	�l��3������)�=�2],���then���E���[�N�@�]�=��O�UV�=��X�N��O�E�as���an��O��-mo�S�dule.���In��fact��	��l���O�UV�=��X�N�@�O����=���UR��f\�(�����d���c��F���̽N��
������F���̽N�����J9��if��N�splits�in��O��������c��F��ؽN���"����2������J9��if��N�do�S�es�not�split�in��O��.�������Џ�The���ab�S�o��rv�e�co�v�ers�all�p�S�ossibilities,��!b�ecause��N��6>�UR�3�can't�ramify�in��O�UV�.�}By�Kummer's�theorem,����N��̹splits�|�in��O��>�i�����f�������Fu��
����3��
���콉fe
�|��'�����N����������f���� �=�UR1,���and�there�are�exactly�t��rw�o�|�O�UV�-stable�submo�S�dules�of��O��=��X�N�@�O��.�KIn�the���second��case,�whic��rh�happ�S�ens�i�����f�������Fu�����3����콉fe
�|��'�����N������&n��f���� t��=�UR��1,��O�UV�=��X�N�@�O�?��has�no��O��-stable�submo�S�dules.�8�Therefore�� J,��B��n���̺0��V�=���UR��f\�(�����d�������Fu�����N��"��1�����콉feu���'���3�����'�[�+���2����C��if�꨼N�+��splits�in��O�UV�,�i.e.,��N��6��UR�1���(�mo�S�d���B3).�������������Fu�����N��"�+1�����콉feu���'���3��������C���if�꨼N�+��do�S�es�not�split�in��O�UV�,�i.e.,��N��6��UR�2���(�mo�d���B3)������������28�����1��K����b�{�K������{��In��the�rst�case,�note�that�the�t��rw�o�꨿O�UV�-stable�submo�S�dules�ha��rv�e��to�b�e�coun��rted�separately��V.�������Similarly��w��re�obtain��!U]���ün���̺1728���b�=���UR��f\�(�����d�������Fu�����N��"��1�����콉feu���'���2�����'�[�+���2����C��if�꨼N��6��UR�1���(�mo�S�d���B4)�������������Fu�����N��"�+1�����콉feu���'���2��������C���if�꨼N��6��UR�3���(�mo�S�d���B4)�������"
�dep�S�ending��on�whether�N�+��splits�in��Q�(�i�).�������Examples.�\p�F��Vor�KؼN�;��=���37�the�gen��rus�is�2.�Using�the�form��rula,�d$w�e�K�get�2�g�[ ���ӹ2���=�36����(2�+���18)������14�UR=�2��(so�it�w��rorks).������Using��the�form��rula�w�e�can�also�obtain��g�n9�(�X���̺0����(13))�UR=�0��and��g��(�X���̺0����(11))�UR=�1.������It�� is�therefore�clear�that�the�gen��rus�of��X���̺0����(�N�@�)�is�appro�ximately��N�F:=�12.�h In�the�article�b�y���Serre�A�in�the�Lecture�Notes�in�Mathematics�349�(An��rt�w�erp),�cbw�e�A� nd�the�follo��rwing�table.��W��Vrite����N��6�=�UR12�a����+��b�꨹with�0����b����11.�8�Then�� T捍���s1�������b�T��Y���ff�����<�1����Y�5������7�����11����o؟��ff�ᗟ &c����g�;�Y���ff����a������1���>�4�a���P�5a���c6a����+�1.������ �e�Hence������ 1���+��g�n9�(�X���̺0����(�p�))�UR=��dim��꘼M���̽p�+1���ٹ(�SL��� ދ��̺2�����(�Z�))�:������It���w��ras�Serre's�idea�to�think�of�\mo�S�dular�forms�mo�d��p�",��for�some�congruence�subgroup��U]��UR�3�����f\� �����d���?��1��� �1�������?�0��� �1�����*����f\�!���2ꢹ,���lik��re�� ���̺0����(�N�@�)�or����̺1���(�N�@�).�4\W��Ve�could�use�our�mo�S�duli�theoretic�in��rterpretations,���but��U^�instead��w��re'll�de ne��������M���̽k��#��(�;����F���̽p���]�)�UR���F���̽p���[[�q�n9�]]�:������By�5Shim��rura's�cohomology�tric�k,� �w�e�kno�w�that��M���̽k��#��(�;����Z�)�is�a�lattice�in��M���̽k���(�;����C�).���Hence���w��re��can�set����� �M���̽k��#��(�;����F���̽p���]�)�UR=��M���̽k���(�;����Z�)���� �����Z�� '��F���̽p���]�:����Then���Serr��ffe's�35e�quality��states�that�for�a�prime��p�,������-W�M���̽p�+1���ٹ(�SL��� ދ��̺2�����(�Z�)�;����F���̽p���]�)�UR=��M���̺2����(���̺0���(�p�)�;��F���̽p���]�)���in�K��F���̽p���]�[[�q�n9�]].��The�philosoph��ry�is��mo��ffd��)�p��forms�with��p��in�the�level�c�an�b�e�taken�to�mo�d��p��formswith����no�.Ҽp��in�the�level,�m�but�of�a�higher�weight�.�nSo��ffor�example��M���̽k��#��(���̺1����(�p����2� �����N�@�)�;����F���̽p���]�)�is�a�subset�of����M���̺?�����(���̺1����(�N�@�)�;����F���̽p���]�)��of�forms�of�some�higher�lev��rel.������Finally��V,�0cconsider��the�map�from�the�righ��rt�hand�side�to�the�left�hand�side�in�Serre's�equalit�y��V.���Recall��that��qʍ��3o�G���̽k��x�=������ō�����B���̽k������Qm�fef �  ���B�2�k����� �k�+�������m �1�� ���t*�����X��� ҁ�����n�=1����r����̽k��#��(�n�)�q��n9�� ��n�� kۿ2�UR�M���̽k���(�SL��� ދ��̺2�����(�Z�))�:�� R���By��Kummer,��ord���〟�̽p���ݹ(�B���̽p��1���ٹ)�UR=���1,��so��[email protected]��h�ݼE���̽p��1���+�=�UR1���+������ō��ۿ�2(�p����1)���۟Qm�fe2�+�  �� �B���̽p��1���������9�7�����X����JK�����̽p��1���ٹ(�n�)�q��n9�� ��n�� kۿ��1� ��(�mo�S�d���B�p�)�:��� �����29���� �!��K����b�{�K������{��Hence�Nw��re�got�the�map�from�the�righ�t�to�the�left:��m�ultiply�b�y��E���̽p��1����to�get�to��M���̽p�+1���ٹ(���̺0����(�p�)�;����F���̽p���]�).����Then���tak��re�the�trace�to�get�to��M���̽p�+1���ٹ(�SL��� ދ��̺2�����(�Z�)�;����F���̽p���]�).�$�The�trace�map�is�dual�to�the�inclusion�and���is��expressed�b��ry��'֍����tr���: (�f�G��)�UR=���UQ���p�+1��
������Z�����X������i�=1���y)�f��j�
���̽i���ʼ
���̽i���,�2�����̺0����(�p�)����n���SL����3��̺2��I7�(�Z�)�:��,	��F��aGebruary�z�14�and�16,�1996��b#��Scrib�S�e:�8�Jessica��P��rolito,��<[email protected]>�������Our���goal,�9zfor�these�t��rw�o���da�ys,�is���to�dene�the�mo�S�dular�curv��res���X��(�N�@�)��(lo�v�er��Q�(����̽N��D�),�9zand����X���̺1����(�N�@�),���and���X���̺0���(�N��)��o��rv�er��Q�.�*These�notes�will�sp�S�ell�out�the�construction�of���X��(�N�@�)��"=,���with�some���discussion�ݶof�the�construction�of�the�other�t��rw�o�ݶt�yp�S�es�of�curv�es.��:The�idea�comes�from�Shim�ura.������Let�Δ�Q�(�t�)�b�S�e�the�function�eld�of��P����2�1����=�Q�,�eand�pic��rk�an�elliptic�curv�e���E���̽j����C�=�Q�(�t�)�with��j��ӹ-in�v��X�arian�t����t�:�����������E���̽j�������:�UR�y��n9����2�����=�4�x�����3��j��������ō�+�27�t���۟Qm�fe*h͟�t������1728�����/yۼx����������ō�+�27�t���۟Qm�fe*h͟�t����1728�������$�(Note���that�the�general�form��rula�for�the��j��ӹ-in�v��X�arian�t�of�a�curv�e��y��n9���2�2�����=�UR4�x����2�3���Ŀ����g���̺2����x����g���̺3�����is����j�%�=���������֐�1728�g���-:��I{�3��wq�2�������qu�fe ʟ���g���>���I{�3��wq�2���t"��27�g���>���I{�2��wq�3�������%ׂ�,����and�+ that��j��ӹ(�E���)�determines�the�isomorphism�class�of�the�giv��ren�curv�e��E��3�o�v�er�the�algebraic���closure��of�the� eld�of�de nition.)������By���substituting�in�a�giv��ren�v��X�alue��j�W��for��t�,���w�e�w�ould�get�a�form�ula�for�an�elliptic�curv�e�o�v�er����Q�(�j��ӹ)��dwith��j��-in��rv��X�arian�t��d�j��;�Bfor��j�>I�=��v0�or�1728,���w��re�could�pic�k�a�diferen�t�form�ula�for���E���̽j���aw�whic�h���w��rould��giv�e�an�isomorphic�curv�e�o�v�er��Q�(�t�),�for�whic�h�that�substitution�w�ould�mak�e�sense.������Notice� �that,�R�in�general,�if�w��re�ha�v�e�an�elliptic�curv�e��E��=K�ܞ�,�R�with��K��H�some� eld�of�c�har-���ictaristic�֌prime�to��N�@�,��then�w��re�can�consider��E���[�N��](�������fe ۶� fb��K���� ۶�),��the�set�of�all��N��-torsion�p�S�oin��rt�of��E����de ned�ao��rv�er��������fe ۶� fb��K������,�c�whic�h�is�isomorphic�to�(�Z�=��X�N�@��Z�)����2�2����.�� Then�w�e�let��K�ܞ�(�E���[�N�@�])�b�S�e�the�smallest���extension���of��K��T�o��rv�er���whic�h�all�the�p�S�oin�ts�of��E���[�N�@�]�are�de ned.�c Notice�also�that��E�=K��T�and��N����together���de ne�a�represen��rtation�of�the�Galois�groups��G��.�al�C��(�������fe ۶� fb��K���� ۶=K�ܞ�)�in�to�the�automorphisms�of���the���N�@�-torsion�p�S�oin��rts�of��E���,��as�they�are�de ned�b�y�p�S�olymial�equations�with�co�ecien��rts�in��E���.���W��Ve��get����r}+����̽E�r�;N��x��:�UR�G��.�al�C��(�������fe ۶� fb��K���� ۶=K�ܞ�)��!���Aut��=(�E���[�N�@�])�����P�������԰���n:�=��������GL��� ����̺2��%O��(�Z�=��X�N��Z�)���with�G��G��.�al�C��(�������fe ۶� fb��K���� ۶=K�ܞ�(�E���[�N�@�]))��q=��k��rer��R(����̽E�r�;N��#@�).� PUnsurprisingly��V,���w��re�will�frequen�tly�lea�v�e�out�the��N�@�,���writing��simply�����̽E��-��.������No��rw,�E=to�3 return�to�our�construction,�where��K��I�=�Ы�Q�(�t�).�FW��Ve�will�sho��rw�that��Q�(�t�)(�E���[�N�@�])�is���the��function� eld�of�a�curv��re�de ned�o�v�er��Q�(����̽N��D�)�(so��Q�(�t�)(�E���[�N�@�])� �\������fe #�� n��Q����e�=�z��Q�(����̽N���)),��.and��that���this��curv��re�corresp�S�onds�to��X��(�N�@�).������W��Ve�C�will�actually�do�this,�Z not�b��ry�lo�S�oking�at�����̽E��-��,�but�instead�at������&�fe <��]ڍ����̽E��������,�where������&�fe <��]ڍ����̽E�����Ą�is�giv��ren�b�y��� ����30���� ����K����b�{�K������{��reducing��the�image�of�����̽E�� 0�mo�S�d���1:��6Э������������G��.�al�C��(�������fe ۶� fb��K���� ۶=K�ܞ�)���� ���Y��feSd����@������r����̽E��������������Y��O� line10�-���������a�GL�����̺2�� ���(�Z�=��X�N�@��Z�)�������Ō���H����0���VH�����0���H����y���V�H���� F��%H�����џ�jH����g\� M�H��� ���H���������� �������4���&�fe <��]ڍ����̽E��������� ���j�����������k�GL����f��̺2���& �(�Z�=��X�N�@��Z�)�=�f�1�g�������������?��������������fe��������.jZ����Note:�8����̽E�� 0�is��surjectiv��re�i ������&�fe <��]ڍ����̽E�����L�is.��������Pr���oof.����If���ꨟ��&�fe <��]ڍ����̽E�����L�is��surjectiv��re,�then�� �s���3��������f\� �����d���ꪹ0��� ʦ��1��������1���#uR0�����2����f\�!���>?��2��UR�im��c�(����̽E��-��)�:�� �t��By��squaring,�w��re�see�that��U]�����2��f\� �����d�����޿�1������0����������0����21��1������g���f\�!����2��UR�im��c�(����̽E��-��)�;��䁍�and��th��rus���1�UR�2���im��c�(����̽E��-��),��so�����̽E�� 0�is�surjectiv�e.������(Ob��rviously��V,��if�����̽E�� 0�is�surjectiv�e,�so�is������&�fe <��]ڍ����̽E�����'��.)��ɚ�������Wh��ry���will�w�e�use������&�fe <��]ڍ����̽E������6�instead�of�����̽E��-��?�#2The�k�ernal�of�����̽E�� �%�is�the� eld�generated�o�v�er��K��;�b�y�the����X�'�and�5��Y���co�S�ordinates�of�the�non-zero�p�oin��rts�of���E���̽j���N3�[�N�@�],�H;whereas�the�k�ernal�of������&�fe <��]ڍ����̽E�������is�the� eld���generated��Gb��ry�just�the��X��ʹco�S�ordinates�of�those�p�oin��rts.�:�This�follo�ws�b�S�ecause��X��(�P��ƹ)�Va=��X��(��P��ƹ),���while�� �Y��p�(�P��ƹ)�UR=���Y��(��P��ƹ),�ܢas�� then�the��X�ʣ�but�not�the��Y�u��co�S�ordinates�are� xed�b��ry���1.�3(W��Ve�are���assuming��that�w��re�ha�v�e�a�W��Veierstrass�mo�S�del�for���E���̽j������here.)���=����Remark��6���?�6�L��ffet�f7�E���̺1��&;�and��E���̺2���b��ffe�el���liptic�curves�over��K�B��with�e�qual��j� �invariants.�"(In�other�wor�ds,����E���̺1�� ���and�߈�E���̺2���ar��ffe�isomorphic�over�������fe ۶� fb��K����>�.)�kaThen�����̽E�� q�1������is�surje�ctive�i �����̽E�� q�2������is.�ka(We'l���l�assume�that���either�35�N��6�6�=�UR2�;����3��or��E��L�do��ffes�not�have�c�omplex�multiplic�ation.)�������Pr���oof.� ��Assume�v����̽E�� q�1���[]�is�surjectiv��re.���Then��E���̺1�� 6 �do�S�es�not�ha�v�e�complex�m�ultiplication�o�v�er���������fe ۶� fb��K��� ۶�.�u�(If�T>�E�U�has�CM,�then�the�image�of�����̽E�� �ƹis�ab�S�elian�for�some�quadratic�extension�of��K�ܞ�,�n�but����GL�������̺2��O��(�Z�=��X�N�@��Z�)��has�no�ab�S�elian�subgroup�of�index�2,�as��N��6>�UR�3.)������Then���Aut����(�E���̺1����)��0=��f�1�g���Pic��rk�an�isomorphism�� ��0�:��E���̺1������P��� L4����԰��� e �=�����-��E���̺2�� � �o�v�er��������fe ۶� fb��K����ι.� �0Then,�F�for��an�y�����Ë�2�URG��.�al�C��(������fe ۶� fb��K��� ۶=K�ܞ�),���w��re���ha�v�e�����2���O� �UR�:�����2����Q�E���̺1������P���V����԰���.>�=�����������2���d��E���̺2����,���and�����2���O�E���̽i���,�=��E���̽i��dڹ.��ITh��rus�����2���O� ��=��� ��for�all���Ë�2�G��.�al�C��(������fe ۶� fb��K��� ۶=K�ܞ�).���So���� �UR�:��E���̺1����[�N�@�]�����P�������԰���n:�=��������E���̺2���[�N�@�]���do�S�es�not�quite�giv��re�us�an�isomorphism�of�represen�tations�����̽E�� q�1���s߹and�����̽E�� q�2��� �W�,���but���it�do�S�es�giv��re�us�an�isomorphism�����&�fe�˟]ڍ����̽E�� q�1���������P���ۿ���԰���!ù=������){����&�fe�˟]ڍ����̽E�� q�2�����:pm�.�\�W��Ve�then�ha�v�e�that�����&�fe�˟]ڍ����̽E�� q�2������k�,�9�and�th�us�����̽E�� q�2��� �W�,�9�is���surjectiv��re.��v�������F��Vrom��here�on,�let��K�1�=�UR�C�(�t�),���E���̽j������and�����̽E��-��,������&�fe <��]ڍ����̽E�����L�as�ab�S�o��rv�e.�8�W�e��ha�v�e�����������̽E�� �ڹ:�UR�G��.�al�C��(�������fe ۶� fb��K���� ۶=K�ܞ�)��!���GL������̺2����(�Z�=��X�N�@��Z�)�;����so�l the�determinan��rt�of�����̽E�� ���is�a�cyclotomic�c�haracter.��As�the��N�@�th�ro�S�ots�of�unit�y�are�all�already����in�꨼K�ܞ�,�this�means�that�the�determinan��rt�of�����̽E�� 0�is�trivial,�so�����"����̽E�� �ڹ:�UR�G��.�al�C��(�������fe ۶� fb��K���� ۶=K�ܞ�)��!���SL���3ݟ�̺2����(�Z�=��X�N�@��Z�)�:��� �����31���� ����K����b�{�K������{��Our��goal�is�to�pro��rv�e��that�����̽E�� 0�is�surjectiv��re�on�to��SL����3��̺2���7�(�Z�=��X�N�@��Z�).�������(F��Vor��an��ry� eld��F�!��of�c�haracteristic�not�dividing��p�,��>there�is�an�algebraic�pro�S�of�b�y�Igusa�that���a���\generic�elliptic�curv��re"�(suc�h�as���E���̽j���&5�here),���has�the�maximal�p�S�ossible�Galois�action�on�its���division��p�S�oin��rts.)������W��Ve�*�will�no��rw�in�tro�S�duce�another�w�a�y�of�lo�S�oking�at��K�ܞ�.���It�is�equal�to��C�(�j��ӹ)��;=��F���̺1����,�:�the�*� eld���of��'mo�S�dular�functions�of�lev��rel�1.� �(A��mo�dular�function�of�lev��rel��N�� �is�a�(�N�@�)-in�v��X�arian�t�function���on����H��whic��rh�is�meromorphic�on��H��and�at�the�cusps.)�f�The�set�of�all�mo�S�dular�functions�for����SL��� ދ��̺2�����(�Z�)���are�generated�o��rv�er����C��b��ry�a�single�function��j��ӹ.�I7Similarly��V,��̿F���̽N�� � �is�the� eld�of�mo�S�dular���functions��of�lev��rel��N�@�.�8�W��Ve�then�ha�v�e�the�to�w�er�of� eld�extensions��/Z������ʍ����΄d�����fe ۶� fb��K��������=�����0B�����fe � fa��F���̺1�����������5��j���������F���̽N��������5��j������΄d�K�����=����0B�F���̺1������5[�����SL��� ~��̺2��$>��(�Z�)�G?acts�on��F���̽N��D�,�g�via�the�map��f�G��(��W�)�UR�!��f���G���f�������Fu��
t�a��r�+�b��
t�콉feuI��'��Ec��r�+�d���������f����%t�,�and�G?xes��F���̺1����.�hBy�reviewing�denitions,���w��re�͓see�that��SL������̺2��l"�(�Z�)�=�(�N�@�)�א=��SL������̺2��v�(�Z�=��X�N��Z�)�͓acts�on��F���̽N��D�,�Nand�that���1�acts�trivially��V.��W�e�͓will���sho��rw���that��SL���oe��̺2��/i�(�Z�=��X�N�@��Z�)�=�f�1�g��is�that�Galois�group�of��F���̽N��D�=�F���̺1����,���b�y�exhibiting�a�set�of�functions���on��whose�p�S�erm��rutations�this�group�acts�faithfully��V.������So,��the�en��rtire�picture�so�far�is:��C������� ����(
�����fe�B�	fa��F���̽N������������ҟ��������ʟ����)UX���i���*������������������������������K�ܞ�(��E���̽j���
��[�N�@�])�=�f�1�g���(
F���̽N������06w���Ή��fe�������������ҟ��Q�����ʟ�*�Q���)UXQ���i���Q��������Q������������������d��F���̺1��V�=�UR�K�����06w���͉��fe��������A�E����(Recall�N�that��K�ܞ�(��E���̽j���
��[�N�@�])�=�f�1�g��is�the�exed�eld�of�the�k��rernal�of������&�fe
<��]ڍ����̽E�������.)�e�W��Ve�will�sho�w�that����K�ܞ�(�E���[�N�@�])�B�=��F���̽N��D�,��at�v&the�same�time�that�w��re�sho�w�that�the�Galois�group�is�what�w�e�w�an�t�it���to��Cb�S�e.�h�T��Vo�do�this,���w��re'll�use�the�W�eierstrass��}�-function�attac��rhed�to�a�giv�en�lattice��L��of��C����(and���th��rus�to�the�corresp�S�onding�elliptic�curv�e��E��=�C�).�*BSo�w�e�will�pic�k�a�sp�S�ecic����o�2�UR�H�.�*B(Then���sending�:üt��to��j��ӹ(��W�)�giv��res�us�a�map�from���E���̽j����Sr��t����to��E���̽��2&�,�N�the�elliptic�curv�e�o�v�er��C��with��j��ӹ-in�v��X�arian�t����j��ӹ(��W�),�s�whic��rh�X[is�also�giv�en�b�y�the�lattice��Z��X�+��Z��W�.)���T��Vo�X[the�lattice��L��=��Z��X�+��Z���x�w�e�X[attac�h��}z���,���whic��rh��is�a�mo�S�dular�form�of�w�eigh�t�2�satisying�the�dieren�tial�equation�������
�}�����0���9�(�z���)�����2��V�=�UR�}�(�z��)�����2��j������4�g���̺2����(��W�)�}�(�z��)�g���̺3���(��W�)�;����whic��rh,��=replacing�Y��}��b�y��X�K�and��}����2�0��'��b�y��Y��p�,��=is�a�W��Veierstrass�equation�for��E���̽��2&�.�	�xTh�us�w�e�ha�v�e�����}�(�����Fu��33�r�<r��r�+�s��33�콉feO<��'����N��������;����Z�4��+��Z��W�),��with���r��r;�s�UR�2��Z���and�not�b�S�oth�divisible�b��ry�N,�running�through�the��X��-co�ordinates���of��p�S�oin��rts�in��E���̽��2&�[�N�@�].�8�This�form�is�of�w�eigh�t�2;�w�e�need�to�nd�mo�S�dular�functions,�so�let��2ԍ�����f���ߺ(�r�Î;s�)����(��W�)�UR:����o�7!������ō����g���̺2����(���)�����Qm�fe�L���g���̺3����(���)�������}�������f^��������ō�
���r�S��Ź+����s��
���Qm�fe ;���
�SN�����+�;����Z����+��Z�����f^����_�P�:��������32����!���K����b�{�K������{��(The�Ҽf�]ѹare�certainly�meromorphic�on��H��and�at�the�cusps,�@cas��g���̺2����;���g���̺3���ֹand��}��all�are.)����SL����y��̺2���}�(�Z�)�acts����on�"-these�functions��f�j,�b��ry����a�:��Ҽf���ߺ(�r�Î;s�)����(����W�)�=��f���ߺ(�r�Î;s�)���v=�(���),�0where�"-��5��simply�m��rultiplies�the�ro�w�v�ector���(�r��r;���s�)�%�on�the�righ�t.��/Then�w�e�can�see�that��f�f���ߺ(�r�Î;s�)����g��is�in�v��X�arian�t�under�the�action�of�[�N�@�],�Mso�they���are���p�S�erm��ruted�b�y��SL����X��̺2���\�(�Z�=��X�N�@��Z�).��In�fact,��as��}��is�an�ev�en�function�(�X��(�P��ƹ)�UR=��X��(��P��ƹ)���for�an��ry�p�S�oin�t���on��the�eliptic�curv��re),��f���ߺ(�r�Î;s�)���N�is�also�xed�b�y���1,�so�they�are�p�S�erm�uted�b�y��SL����3��̺2���7�(�Z�=��X�N�@��Z�)�=�f�1�g�.������W��Ve�'Yha��rv�e�dened��f���̽r�Î;s�����for��r�;���s�p^�2��Z�;�,�v�not�b�S�oth�divisible�b��ry�N.�In�fact,�if�(�r��r;���s�)�p^=�(�r��S����2�0��!Ǽ;�s����2�0���9�)���(�mo�S�d���B�N�@�),���then����f���ߺ(�r�Î;s�)��c{�=�bռf���ߺ(�r��<r�����0���9�;s������0���Ǻ)����,�so�w��re�only�need�to�consider�(�r�;���s�)�bտ2��Z�=��X�N�@��Z����2�2����=�f�1�gn�(0�;��0)���As����f���ߺ(�r�Î;s�)����(��W�)�wruns�o��rv�er�wall�of�the��X��-co�S�ordinates�of�p�oin��rts�in��E���̽��2&�[�N�@�]�whic�h�are�not�zero�for�these���(�r��r;���s�),�-w�e���m�ust�ha�v�e�these��f���ߺ(�r�Î;s�)���all�distinct.��(There�are��N��@���2�2��	R���Q��1�non-zero�p�S�oin�ts�on��E���[�N�@�],���and����X��(�P��ƹ)���=��X��(�Q�)���i��P����=��Ŀ�Q�,��so�there�are�exactly�as�man��ry�pairs�(�r�;���s�)�as�there�are��X����co�S�ordinates��of�p�oin��rts�in��E���[�N�@�].)������Th��rus�Sqw�e�ha�v�e�sho�wn�that�the�non-zero�p�S�oin�ts�of���E���̽j���l �[�N�@�]�ha�v�e��X��-co�S�ordinates�(in�������fe�B�	fa��F���̽N������)�giv�en���b��ry��the��f���ߺ(�r�Î;s�)��U��2�URF���̽N��D�,�and�th�us��F���̽N��n��=�UR�K�ܞ�(��E���̽j���
��[�N�@�]).������(As�8%a�corollary��V,�[�w��re�ha�v�e�sho�wn�that�the�Galois�group�of��F���̽N��Qi�o�v�er��F���̺1���)�is��SL������̺2��ִ�(�Z�=��X�N�@��Z�)�=�f�1�g�,���as��this�group�acts�faithfullly�on�the��f���ߺ(�r�Î;s�)����.)������Th��rus,�mnally��V,�w�e��Eha�v�e�sho�wn�that�for�an�y��E��=�C�(�t�)�with��j��ӹ-in�v��X�arian�t��t�,�mthe�asso�S�ciated���represen��rtation�꨼���̽E��0�has�image��SL����3��̺2���7�(�Z�=��X�N�@��Z�).������W��Ve�g�kno��rw�that,��Bif�w�e�let��K�/�=�*��Q�(����̽N��D�)(�t�),��B�E��an�elliptic�curv�e�with��j��ӹ-in�v��X�arian�t��t�,��Bthen�����̽E�����has�"�image�con��rtained�in��S���L���̺2����(�Z�=��X�N�@��Z�).��(This�do�S�esn't�w�ork�for��K��$�=����Q�(�t�),�0�as�w�e�need�to�ha�v�e�������̽n�� �Z��� �K����in��order�to�ha��rv�e��the�image�of�����̽E�� ��con��rtained�in��S���L���̺2����(�Z�=��X�N�@��Z�).)��%F��Vurthermore,��if�w�e���replace���C�(�t�)�b��ry��Q�(����̽N��D�)(�t�),�the�image�of�����̽E�� 0�can�only�get�larger.�8�In�the�follo�wing�diagram���D����� ����W�C�(�t�)(�E���[�N�@�])���������� �������E��������x�UX�����p��*�����h���������������G����2�0���������������X���Q�����P��*�Q����H�UXQ���[email protected]��Q��� �8� ��Q����������������������C�(�t�)����}��Q�(����̽N��D�)(�t�)(�E���[�N�@�])��������������Q����E���*�Q�����x�UXQ�����p��Q����h� ��Q�������������������X� �������P������H�UX���[email protected]��*���� �8�������������� �G���������������s�Q�(����̽N��D�)(�t�)������G��G����and��G����2�0���ܹare�the�Galois�groups�(and�also�the�images�of�����̽E�� *+�o��rv�er����Q�(����̽N��D�)(�t�)�and��C�(�t�)�resp�S�ec-���tiv��rely).�CBasic�CdGalois�theory�tells�us�that��G����2�0�������\�G�,�Y�but�w�e�also�kno�w�that��G����2�0�����=���\SL������̺2����(�Z�=��X�N�@��Z�)���and�꨼G�UR����SL���3ݟ�̺2����(�Z�=��X�N�@��Z�),�so��G��=��G����2�0��#��=��SL���3ݟ�̺2����(�Z�=��X�N�@��Z�).������Next,��0let�rus�consider��E��=�Q�(�t�),�as�alw��ra�ys�rwith��j��ӹ-in��rv��X�arian�t�r�t�.��Then�����̽E�� �ڹ:�UR�G��.�al�C��(�Q�(�t�)(�E���[�N�@�])��!�����GL�������̺2��O��(�Z�=��X�N�@��Z�).��4The�!determinan��rt�of�����̽E�� N��is�again�a�cyclotomic�c�haracter,�.�but�is�not�trivial�this���time;���indeed,�ݺit��is�surjectiv��re�and�th�us�con�tains�a�set�of�coset�represen�tativ�es�for��SL���� ��̺2��y�(�Z�=��X�N�@��Z�)���in����GL���OP��̺2��T�(�Z�=��X�N�@��Z�).��W��Ve���see�that�����̽E�� �=�is�again�surjectiv��re,���although�on�to��GL���OP��̺2��T�(�Z�=��X�N�@��Z�)�this�time.��� ����33����"�~��K����b�{�K������{��Notice��that,�in�this�case,��Q�(�t�)(�E���[�N�@�])�con��rtains�����̽N��D�,�so�w�e�no�w�ha�v�e�the�diagram��j������0����������fe #�� n��Q����� ��Q�(�t�)(�E���[�N�@�])���.�����=�Q�(����̽N��D�)������������fe�����B��Q�(����̽N��D�)(�t�)�����*�������fe����������&���ܬS����(��$�S����*~�|TS����,K��S����.�+�S����/�'��S����1��0�DS����3�:3S�����펍�?󎎎�����jF�Q�(�t�)�����*�������fe������������m����������������UX���,���*����l��������������������ߛU�Q������_�k��Basic��Galois�theory�no��rw�tells�us�that������fe
#��	n��Q����ݿ\����Q�(�t�)(�E���[�N�@�])�UR=��Q�(����̽N��D�):��������Pr���oof.����Let��ѼF����b�S�e�the�in��rtersection�of�these�t�w�o�elds.�E\It�clearly�con�tains��Q�(����̽N��D�).�E\If��F����is���bigger��than��Q�(����̽N��D�),�then��F��ƹ(�t�)�is�bigger�than��Q�(����̽N���)(�t�),�and�th��rus��.�����K�G��.�al�C��(�Q�(�t�)(�E���[�N�@�])�=F��ƹ(�t�))�UR�(���SL���3ݟ�̺2����(�Z�=��X�N��Z�)�:����But��w��re�kno�w�(as�ab�S�o�v�e)�that�����Lu�SL���ZTZ��̺2��_^�(�Z�=��X�N�@��Z�)�UR=��G��.�al�C��(�C�(�t�)(�E���[�N��]�=�C�(�t�))���G��.�al�C��(�Q�(�t�)(�E���[�N��])�=F��ƹ(�t�))���so��w��re�m�ust�ha�v�e�equalit�y�ab�S�o�v�e,�and�th�us��F���=�UR�Q�(����̽N��D�).�������Recall�a�that��C�(�t�)(�E���[�N�@�])�=�C�(�t�)�is�the�function�eld�for��X��(�N��).���W��Ve�ha��rv�e�a�sho�wn�that����Q�(�t�)(�E���[�N�@�])�]�is�a�function�eld�for�the�corresp�S�onding�curv��re�dened�o�v�er��Q�(����̽N��D�),�z>so�w�e�ha�v�e���dened�꨼X��(�N�@�)�o��rv�er���Q�(����̽N��D�).������W��Ve�V~can�similarly�dene��X���̺1����(�N�@�)�and��X���̺0���(�N�@�)�o��rv�er�V~�Q�.�}Let��L�UR�=��K�ܞ�(�E���[�N��])�V~(where��Q�(�t�)�UR=��K�ܞ�).���Then��W�G��.�al�C��(�L=K�ܞ�)�UR=��GL������̺2����(�Z�=��X�N�@��Z�).�pConsider�the�subgroup��H�v��of�this�Galois�group�consisting�of��	�all��matrices�of�the�form�����f\� �����d����ʿ��� ����������͹0��� �˿�����+�̟�f\�!���3�x�.�2Then��L����2�H��
���\��������fe
#��	n��Q������=�UR�Q�(����̽N��D�)����2�H��n��=��Q�.�As���L����2�H���b�is�an�extension�of��U^��Q�	ιof�transcendence�degree�1,��it�is�the�function�eld�of�a�curv��re�dened�o�v�er��Q�.��QIt�turns�out��U]�to��b�S�e�the�curv��re��X���̺0����(�N�@�).�8�T��Vo�get��X���̺1���(�N�@�),�w��re�w�ould�use�the�subgroup��H�B��=���UR��f\� �����d���?����� ?��������P�0��� P1�����[email protected]��f\�!���3*��.��+	Z��21�z�F��aGebruary�,�1996��b#��Scrib�S�e:�8�Da��rvid��M�Jones,��<[email protected]>��Qx����First�p�of�all,��(w��re�start�with�a�correction�from�last�time.���Let��L=K�MF�b�S�e�a�Galois�extension.���Let�꨼E���߽=L��r��b�S�e�an�elliptic�curv��re.�8�Supp�ose�that��j��ӹ(�E���)�UR�2��K��F�and��for�all��g�Ë�2�URG��.�al�C��(�L=K�ܞ�),�����2�g�����E�����P���	i����԰���"Q�=������
^���߽=L��;�E��.������Last���time�it�w��ras�stated�that�w�e�could�conclude�that�there�exists���E���̺0����V��L̽=K���7�suc�h�that��E�����P���	i����԰���"Q�=������
^���߽=L��;�E���̺0����.���This��is�not�correct.�8�W��Ve�ma��ry�only�conclude�that��E�����P���	i����԰���"Q�=������
^��zP�=����P��\������L����;�E���̺0����.�The�p�S�oin��rt�is�that��L�UR�=����x��zK������K���1�.�������34����#�L��K����b�{�K������{����Let�,	�E���߽=K���Ϲb�S�e�an�elliptic�curv��re.��Let��g�撿2�xYG��.�al�C��(�L=K�ܞ�)�for�some�extension��L=K��.��Then�let�� \�����̽g��*P�:��UR���2�g��*R�E����K���o����g���
�D�����������.���g	��	i�!��������^��E���b�S�e�^�a�family�of�isomorphisms.�
0No��rw,�z�����̽g�I{h��
��and�����̽g��a���������2�g��a�����̽h��ĉ�are�b�oth�isomorphisms�from��������2�g�I{h����E����to�꨼E��so�they�dier�b��ry�an�elemen�t�of�Aut(�E���)�UR=��f�1�g�.��������Denition��8���K���L��ffet���g�n9;���h�UR�2�G��.�al�C��(�L=K�ܞ�)�.�CZWe�dene��c�(�g�n9;���h�)��2��A��2ut�(�E���)��by�the�r��ffelation��c�(�g�n9;���h�)����̽g�I{h��
A�=�������̽g�����������2�g�������̽h��e��.������Claim��2���4/��c�(�g�n9;���h�)�35�is�a�2-c��ffo�cyle.����Pr���oof.��t�First,��let's�*rewrite��c�(�g�n9;���h�)��[=�����̽g���v����x���2�g���x����̽h��'i����x����̽g�I{h����W
���W��1�����.��fThen�*in�order�to�pro��rv�e�*that��c�(�g�;���h�)�is���a��2-co�S�cycle,�w��re�m�ust�sho�w�the�follo�wing�(see�Serre's�\Lo�S�cal�Fields",�p.113):��������i9����g���>9�c�(�g��n9����0��<r�;���g��n9����00�����)������c�(�g�n9;�g������0��<r�g������00�����)�UR=��c�(�g�n9;�g������0��<r�)������c�(�g�n9g������0���;���g������00�����)���Then�Ss�c�(�g�n9;���g�����2�0��<r�)�uϿ��c�(�g�n9g�����2�0���;���g�����2�00�����)�UR=�����̽g��JͿ��uϟ��2�g��Jϼ���̽g��I{�����0��������uϼ���̽g�I{g�������0�����,8���W��1�� ����uϼ���̽g�I{g�������0���
Q�������2�g�I{g����-:�0���Q�����̽g��I{�����00���.[�������̽g�I{g�������0���B�g�������00�����dğ��W��1��,��=�UR����̽g��������2�g��Jϼ���̽g��I{�����0�����������2�g�I{g����-:�0���Q�����̽g��I{�����00���.[�������̽g�I{g�������0���B�g�������00�����dğ��W��1��,��=��������̽g���'����)���2�g���)�(����̽g��I{�����0���	J����)���2�g��I{��-:�0���J����̽g��I{�����00���	���)��)�������̽g�I{g�������0���B�g�������00���������W��1��,d�=�UR����̽g��������2�g���)�(�c�(�g��n9���2�0��<r�;���g��n9���2�00�����)�������̽g��I{�����0���B�g��I{�����00����Q�)��������̽g�I{g�������0���B�g�������00���������W��1��,d�=��UR���2�g��*R�c�(�g��n9���2�0���;���g��n9���2�00�����)�������̽g���'������2�g���)����̽g��I{�����0���B�g��I{�����00����z�������̽g�I{g�������0���B�g�������00���������W��1��,d�=�������2�g����c�(�g��n9���2�0��<r�;���g��n9���2�00�����)������c�(�g�n9;�g�����2�0��<r�g�����2�00�����).�h�V�������W��Ve��w��ran�t�this�2-co�S�cyle��c�(�g�n9;���h�)�to�b�e�trivial.������Lemma��1���<uQ�L��ffet�35�G�UR�=��G��.�al�C��(�L=K�ܞ�)�.�fiIf��L��=����x��zK������K���1�,�then��H���V���2�2���Z�(�G;����f�1�g�)��!��H���V���2�2���(�G;���L����2�����)�35�is�inje��ffctive.����Pr���oof.����Consider��the�exact�sequence��������-�0�UR�!�f�1�g�!����x��zK�������K��ܞ���������������
���-:�2���g	��F^�!�������x��(��������%���K��ܞ��������8���!��0�:������Lo�S�oking��at�a�piece�of�the�long�exact�cohomology�sequence�w��re�get����]�M�:::�UR�!��H���V����1���Z�(�G;����x��$����������K��ܞ�� ���������)��!��H���V�� ��2���(�G;����f�1�g�)��!��H���V�� ��2���(�G;����x��$����������K��ܞ�����������)[2]��!��0�:����No��rw�>�H���V���2�1���Z�(�G;����x��$����������K��ܞ���2��������)��d=�0�b��ry�Hilb�S�ert�Theorem�90�so��H���V���2�2���Z�(�G;����f�1�g�)�is�isomorphic�to�the�2-torsion����in��S�H���V���2�2���Z�(�G;����x��$����������K��ܞ���2��������).�/�So��H���V���2�2���(�G;����f�1�g�)�certainly�injects�in��rto��H���V���2�2���(�G;����x��$����������K��ܞ���2��������).�/�Since��L�r��=����x����������K���Nl�,��=the��Sresult���follo��rws.��~����Y����In��order�to�sho��rw�that�the�class�of��c�(�g�n9;���h�)�in��H���V���2�2���Z�(�G;��f�1�g�)�is�trivial,��+b��ry�the�ab�S�o�v�e�lemma,���it��is�enough�to�sho��rw�that�the�class�of��c�(�g�n9;���h�)�in��H���V���2�2���Z�(�G;����x��$��������K��ܞ���2��������)�is�trivial.�8�T��Vo�do�this�w��re�will�use������Dieren��rtials:��kPic�k��a�non-zero�dieren��rtial��!����6�=�OP0�on��E���.���That�is,�^?�!��2�OP�H���V���2�0���Z�(�E��;����
����2�1����).���Let����g���2�j�G��.�al�C��(�L=K�ܞ�).�
�aThen��~����̽g��?ѹ:�����2�g��	?ӼE���������e����g	����!�������E�n��so�the�pullbac��rk������2����RA��g���?ѹ:��H���V���2�0���Z�(�E��;����
����2�1����)��������������g	��!������H���V���2�0���(����2�g����E�;����
����2�1����).�
�aTh��rus,��������2����RA��g������!�Ë�2�UR�H���V���2�0���Z�(����2�g����E��;����
����2�1����),��whic��rh�has�basis�����2�g�����!�X�so�w�e�m�ust�ha�v�e������2����RA��g������!�Ë�=�UR�a���̽g�����������2�g����!�X�for�some��a���̽g��*P�2��L����2�����.������Note��1���-��What�tis�����2�g��I�!�n9�?�)�L��ffet��E���߽=L���<�b�e�an�el���liptic�curve�and�let��L��,���!��M��D�b�e�tan�inje�ction�of�elds.���Then�g.�H���V���2�0���Z�(�E��;����
����2�1����)��'�
���̽L��	��M��u�=����H���V���2�0���(�E���߽=��M��
K�;��
����2�1����)�.�UIn�g.our�c��ffase,�t-�M��u�=����L��and�the�map��L�,���!��M���is�given���by��the�action�of��g����2�KWG��.�al�C��(�L=K�ܞ�)�.���So�we�get��H���V���2�0���Z�(�E��;����
����2�1����)�
�
���̽L��	T�L��=��H���V���2�0���(����2�g����E��;����
����2�1����)�.���The��element�of����H���V���2�0���Z�(����2�g����E��;����
����2�1����)�35�which�c��fforr�esp�onds�35to��!��
����1��under�this�map�is�����2�g��5�!�n9�.��������35����$����K����b�{�K������{����Lemma��2���<uQ�Show�35that��c�(�g�n9;���h�)�UR=�(����2�g����a���̽h��������a���̽g�����)�=a���̽g�I{h�� ���.�����Pr���oof.����First,��consider�the�follo��rwing:�������a���̽g��k���������2�g��k��a���̽h���w������a���̽g�I{h����|v���2��1�� ��!�Ë�=�UR�a���̽g����������2�g���a���̽h���w�����((�����2����RA��g�I{h��� ��)����2��1��� \|���2�g�I{h��m�!�n9�)�UR=��a���̽g��k����(�����2����RA��g�I{h��� ��)����2��1�� �������2�g��k��(�a���̽h���w������2�h���y�!�n9�)�=��a���̽g��k����(�����2����RA��g�I{h��� ��)����2��1�� �������2�g��k��(�����2����RA��h���e�!�n9�)�=���(�����2����RA��g�I{h��� ��)����2��1��$��������2�g����(�����2����RA��h���e�)������(�a���̽g��������2g�����!�n9�)�UR=�(�����2����RA��g�I{h����)����2��1��$��������2�g����(�����2����RA��h���e�)����������2����RA��g������!�Ë�=�UR((����̽g�I{h���)����2��1��$������2�g����(����̽h��e�)�������̽g�����)����2�����!�n9�.���(����But�KA�c�(�g�n9;���h�)���=�����̽g���k����m���2�g���m����̽h��R^����m����̽g�I{h��������W��1��#)��is�m��rultiplication�b�y�a�constan�t�(��1)�so�the�dual�is�m�ulti-���plication���b��ry�the�same�constan�t.�c;Th�us,��Nw�e�ha�v�e��a���̽g���C����E���2�g���E�a���̽h��6����E�a���̽g�I{h�����5���2��1�����!�ۓ�=�mZ�c�(�g�n9;���h�)�!�f��and�as��!��6�=�mZ0,��Nw��re���m��rust��ha�v�e��c�(�g�n9;���h�)�UR=�(�a���̽g�����������2�g����a���̽h��e�)�=a���̽g�I{h��	��.��T�������This��sho��rws�that��c�(�g�n9;���h�)�is�a�cob�S�oundary��V,�and�th�us�its�class�is�trivial�in��H���V���2�2���Z�(�G;����x��$����������K��ܞ���2��������).��2�����Let�6U�N���>���3�b�S�e�an�in��rteger.� �Recall�that�w�e�wrote�do�wn�an�elliptic�curv�e��E���߽=�Q�(�j�v�)���Q�whose����j��ӹ-in��rv��X�arian�t��w�as��j��ӹ(�E���)�UR=��j��.������Let�-��F���̽N�� FԹb�S�e�the� eld�of�mo�dular�functions�of�lev��rel��N�@�.��That�is,�>J�F���̽N�� �y�=��5�Q�(�j��ӹ)(�E���[�N��]�=�f�1�g�).���This�� eld�lies�o��rv�er�꨿F���̺1��V�=�UR�Q�(�j��ӹ).�8�W��Ve�ha��rv�e�꨿G��.�al�C��(�F���̽N��D�=�F���̺1����)�=��GL������̺2����(�Z�=��X�N�@��Z�)�=�f�1�g�.������No��rw�6�de ne��F�� �=����[���̽N��D�F���̽N���.� �This�6� eld,��whic�h�is�the�comp�S�ositum�of�all�the�of�the��F���̽N��D�'s,���corresp�S�onds��to�a�pro�� jectiv��re�system�of�mo�dular�curv��res.������De nition��9���K���The�35 nite�adele�ring�of��Q�,�denote��ffd��A���̽f��w �,�is�������q�f�(�x���̽p���]�)�UR�2���������Y��� ���3�p��� n�Q���̽p�� ��:��x���̽p���2��Z���̽p�����for��almost�all��P�Q�p�g��&���W��Ve��ma��ry�also�describ�S�e��A���̽f�� aǹas���� q^�����Q���c��=�����T^����UR�Z���?�� ����Q�.���j����Notice��that��GL���zC��̺2��:G�(����0^����Z���@�)�UR����GL������̺2����(�A���̽f��w �).������Claim��3���4/��The�35gr��ffoup���GL����П�̺2���Թ(�A���̽f��w �)��acts�on��F�1�.������First,���recall���that��F��c�=�UR�Q�(�f���ߺ(�r�Î;s�)��U��:�(�r��r;���s�)��2��(�Z�=��X�N�@��Z�)����2�2����n�(0�;��0)�;�N��6���1).�� (GL����ß�̺2��kǹ(�Z�=��X�N�@��Z�)���acts�on�the���set��of��f���ߺ(�r�Î;s�)���N�via���荍���Ǟ��f\� �����d�����J�a����kWb��������@�c�����Kd�������M��f\�!����8K�:�UR�f���ߺ(�r�Î;s�)��U��7!��f���ߺ(�ar�<r�+�cs;br��+�ds�)���(� ����Consider�(the�ob�� ject����lim����S � ������T�������T�X��(�N�@�).��_It�is�not�a�v��X�ariet��ry�but�do�S�es�exist�in�some�appropriate���category��V.� �W�e�i�ha��rv�e�the�follo�wing�t�w�o�notions�of�\p�S�oin�ts"�on����lim������� ������T����������X��(�N�@�),���whic�h�are�con�tra-dual��UT�to��eac��rh�other:������1.�8�P��rairs��(�E��;����)�where��E����is�an�elliptic�curv�e�and���UR�:�(�Q�=�Z�)����2�2����������q����g ����!�������E���̺tors�������2.� �ZP��rairs�{(�E��;��� ���)�where��E�ɒ�is�an�elliptic�curv�e�and�� �B�:����5�^������Z����2�2�������������|���g ����!����� ��T���a�(�E���),��0where��T�a�(�E���)��=������lim����O�� ������T�������Q��E���[�N�@�]�UR=�������Q��� ����p���v�T���a���̽p���]�(�E��)�� ����W��Ve��will�no��rw�do�a�w�arm-up�to�sho�w�that��GL���zC��̺2��:G�(�A���̽f��w �)�acts�on��F�1�.��� ����36����%^��K����b�{�K�����������Let���g�� �=���z��f\� �����d���e��a���! �b���������c��� ��d�����+����f\�!���7#�2��z�GL�����x�� �+������ 2���&��(�Q�).�{Lo�S�ok�at�pairs�(�E��;��� ���)�with��E���߽=�C��Ԭ�and�� ��s�:�z��Z����2�2����������q����g ����!�������H���̺1����(�E��(�C�)�;����Z�).��U^�Note��that�w��re�ha�v�e��&Í���fd���{O��Z����2�2�����������������g ���AJ�!��������AL�H���̺1����(�E���(�C�)�;����Z�)��������z�,�����T���O��j�����"ğ����T���x�j����Í��z]��Q����2�2�����������������g ���AJ�!���������J�H���̺1����(�E��;����Q�)�����=�UR�H���̺1����(�E���(�C�)�;����Z�)���� ��Q�����*s鍹W��Ve��denote�this�b�S�ottom�map�� �h�:�UR�Q����2�2����������q����g ����!�������H���̺1����(�E��;����Q�)�as�w��rell.�������Then��w��re� nd��Í���fd����sG� ��7�����g�Ë�:�������Q����2�2�������������$���g	����_�!���������a�H���̺1����(�E��;����Q�)���������C�����S�����셟���S�����Í�����Z����2�2�������������$���g ����_�!��������ͼL����2�0������$	c��where�꨼L����2�0����is�dened�as�the�image�of��Z����2�2�����under����7�����g�n9�.������As�{�w��re�ha�v�e�a�new�lattice��L����2�0���9�,��5it�determines�a�new�elliptic�curv�e��E������2�0���P�.��W��Ve�also�get�a�map�����UR�:��L����2�0�������������g	���9�!������;�H���̺1����(�E������2�0���P�;����Z�).������So�꨼g�Ë�:�UR(�E��;������)��7!��(�E�����2�0���P�;���������2�0���ȹ)��where��������2�0���p�is�the�comp�S�osed�map����������7���g�Ë�:�UR�Z����2�2����������q����g	����!�������L����2�0�������������g	���9�!������;�H���̺1����(�E�����2�0���P�;����Z�).�������Let��ü��ؿ2����H�;���E�Dҹ=��E���̽��2&�;�L���̽����=��Z����+��Z������P����ؿ���԰�����=��������Z����2�2����.�
�1Let���us�denote�the�in��rv�erse���to�this�latter���isomorphism��b��ry�����̽���x�:�UR�Z����2�2����������q����g	����!�������Z����+��Z��W�.�8�Under�꨼���̽��2&�,�(1�;����0)��7!���AŹand�(0�;����1)��7!��1.�������Claim��4���4/��Che��ffck�35that��g�Ë�:�UR(�E���̽��2&�;������̽���)��7!��(�E���̽���r�����0�����;������̽���r�����0����)�35�wher��ffe����W���2�0��z��=�����Fu�����a��r�+�b�����콉feuI��'��Ec��r�+�d������S�=��g�n9�W�.��@v��Pr���oof.����First��of�all,�����̽���r�����0���Ε�tak��res�(1�;����0)�to����W���2�0��z��=�����Fu���a��r�+�b���콉feuI��'��Ec��r�+�d�������and�tak��res�(0�;����1)�to�1.��֒����No��rw��w�e�lo�S�ok�at����������7���g�n9�.��8�GL�����x���{�+�������{2�����(�Q�)�acts�on�the�righ��rt�on��Z����2�2�����so��g�X�sends�(1�;����0)�to��"	�����(1�;����0)���������f\� �����d����T�a��� Nab�������#�c����Ud�����*�W��f\�!���6U�=�UR(�a;�b�)��"
�and�asends�(0�;����1)�to�(�c;�d�).�Next���t��sends�(�a;�b�)�to��a��ƹ+����b�UR�=�(�c��+����d�)���W���2�0���l�and�asends�(�c;���d�)�to��c��+����d�.���No��rw�����just�scales�the�result�so�that�it�is�of�the�form����33�something��4�v.�&That�is,��c(0�;����1)�gets�sen�t�to�����1.�7gThis��<is�dividing�b��ry��c����+����d��and�w�e�see�that�under����������/���g�n9�,��(0�;����1)��<is�sen�t�to�1�and�(1�;����0)�is���sen��rt��to����W���2�0��%V�.��
����2�1����)�where��X��+�is�the�mo�S�dular�curv�e.������No��rw���w�e�are�setting�out�to�dene�the�Hec�k�e�op�S�erators�on�these�mo�dular�forms�{�to�dene���the��Hec��rk�e�op�S�erators�and�Hec�k�e�algebras�in�a�more�algebraic�(or�\arithmetic")�w�a�y��V.�������Recall:�8�F��Vor�꨼N��@���2�0���j�N�@�,�w��re�had�a�to�w�er:��/������&e����뙼X��(�N�@�)����*��F���̽N���������+�#����F"������Ƅ|�X��(�N��@���2�0���)�����¿F���̽N���"�����0���������+�#����F"�����/UR��And��w��re�also�ha�v�e��F��c�=����URlim��������F�����	�!������F���̽N��D�.�8�That�is,��F���is�the�comp�S�ositum�of�the�elds��F���̽N���.������In��rtuitiv�ely��V,����F�1�,�whic��rh�w�is�a�eld,���corresp�S�onds�to�the�\pro-curv�e"����lim����Ǣ� ������T��������h�X��(�N�@�),���whic�h�is�not�an��UT�algebraic��curv��re.������W��Ve���w��rere�discussing�the�op�S�eration�of��GL���5N��̺2���R�(�A���̽f��w�),��}where��A���̽f��	ҹis�the�ring�of�nite�adeles�of��Q�.���j��A���̽f���q�=�����T^����UR�Z���?��
����Q�꨹and��A���̽f��	aǹcon��rtains�����^�����Z���P�as�a�subring.�������Consider��a�pair�(�E��;������),�where��E���߽=�Q��"�is�an�elliptic�curv��re�and��DD���>��h�:�����T^����UR�Z�����2���������������g	��UV�!��������UV�����Y�����"�7�p���-r�T���a���̽p���]�(�E���)��#�=�W��Ve��refer�to�������Q���
\q���p��#̼T���a���̽p���]�(�E���)�as��T�a�(�E���),�the�T��Vate�mo�S�dule�of��E��.��t�����No��rw,��[�T���a�(�E���)�<is�free�of�rank�2�o�v�er����l^�����Z����2�2����<�.�	,�Let��V��p�(�E���)���=��T���a�(�E��)��Y�
�����Z��

��Q��=�������Q���U���p��̰�V���̽p���]�(�E��)�<where����V���̽p���]�(�E���)�UR=��T���a���̽P��̹(�E��)����
�����Z���p���
���Q���̽p���]�.��]܍���Since�e��y��maps������^�����Z����2�2������to��T���a�(�E���),��}w��re�ma�y�extend�b�y��Q��to�get�a�map�(also�called�����)������^�����Q����2�2�������������H���g	��I��!�����I��V��p�(�E���).������If�@A�g���2���ùGL���*^��̺2���b�(�A���̽f��w�),���then��g��:������^�������Q����2�2�����!�������^�������Q����2�2����~T�.�	9�So�b��ry�comp�S�osing���Sйand��g�n9�,���w�e�get�a�new�map�������!�^����Q����2�2�������������V���g	��㑿!�����㓼V��p�(�E���).������Lo�S�ok���at�the�subring�����^�����Z����2�2�����ܹin�����^�����Q����2�2�����}�,��Eand�tak��re�its�image�under����?�����g�n9�,�and�w��re�get�a�lattice��T���Ɵ��2�0��N�in����V��p�(�E���)��whic��rh�is�dieren�t�from��T���a�(�E���).�8�The�follo�wing�picture�illustrates�the�idea:��;�E����z���򺳼T���a�(�E���)����������d�����T��׹�j������������R�^�����̉�Q����2�2����������.��ڰ���g��
ҍ����������g+���g	��۵f�!�����������W��V��p�(�E���)��������h������S��̾Q�j������d�����S��׹�j������������R�^����¾P�Z����2�2���������������g+���g	��۵f�!���������9�T���Ɵ��2�0������������38����'(���K����b�{�K������{��Our�S$goal,�qras�stated�earlier,�is�to�mak��re�the�Hec�k�e�op�S�erators��T���̽n���t�and��<�URd�>�S$�in�to�ob��jects�dened����o��rv�er���Q�.������The��k��rey�idea�is�that��X���̺1����(�N�@�)�and��X���̺0���(�N�@�)�ha��rv�e��Hec�k�e�op�S�erators�acing�as�corresp�ondences.�������Denition��10���Ra��A��.c��fforr�esp�ondenc�e��Db�etwe�en�two�curves��X�z��and��Y�%��is�a�thir�d�curve��C�e��and�two���maps�35��h�:�UR�C�1�!��X�$��and�� ����:��C�1�!��Y��p�.����Giv��ren��a�corresp�S�ondence�(�C�5�;��� ��;� ��O�)��of�curv�es��X��+�and��Y��p�,�w�e�can�form�the�maps������ �� ����� ����(�:�UR�H���V�� ��0���Z�(�X�Jg;���� ��� ��1����)��!��H���V�� ��0���(�C�5�;���� ��� ��1����)���and������)S� �������V�:�UR�H���V�� ��0���Z�(�C�5�;���� ��� ��1����)��!��H���V�� ��0���(�Y�� ;���� ��� ��1����)�;����this��latter�map�� ����������b�S�eing�a�trace�map�of�sorts.������ ������2��� ɥ�is��the�standard�pull-bac��rk�map.�[ � ������� ��can�b�S�e�de ned�using�Serre�dualit�y��V.�[ Recall�that����H���V���2�0���Z�(�C�5�;���� ����2�1����)����2��� m��=����H���V���2�1���(�C�;����O���̽C�� ��)��"and��H���V���2�0���Z�(�Y�� ;�� ����2�1����)����2��� m��=����H���V���2�1���(�Y�� ;��O���̽Y��P��).��NSo��"the�dual�of�� ������� u&�(whic��rh�ma�y�b�S�e���describ�S�ed��as�(� ���������)����2������if�one�lik��res)�ma�y�b�S�e�de ned�via�the�pull-bac�k������Jȼ ���O�� �������:�UR�H���V�� ��1���Z�(�Y�� ;����O���̽Y��P��)��!��H���V�� ��1���(�C�5�;����O���̽C�� ��)�:��&�����W��Ve��will�mak��re�this�explicit�for��T���̽p���]�,��p��not�dividing��N�@�.������In���order�to�de ne��T���̽p��O �(for��p��not�dividing��N�@�)�as�a�corresp�S�ondence�from��X���̺0����(�N��)�to�itself,��zw��re���will��sa��ry�what��C��F�is�and�what�� ��7�and�� ����are.������Let�꨼C�1�=�UR�X���̺0����(�pN�@�).�8�View�����f��X���̺0����(�N�@�)�UR=��f��pairs�� Ɇ(�E��;���C�ܞ�)�j�C�����P���1����԰���Jع=������ܙ�Z�=��X�N��Z��1?�g����b�ȼX���̺0����(�pN�@�)�UR=��f��triples��#��(�E��;���C�5�;�D�S��)�j�C�����P���1����԰���Jع=������ܙ�Z�=��X�N��Z��1?�;�D�����P��������԰����ȹ=������S��Z�=p�Z��,�2�g����Then��w��re�can�consider�the�maps�� ��7�and�� ����as�giv�en�b�y:�����b�� �h�:�UR(�E��;���C�5�;�D�S��)��7!��(�E��;�C�ܞ�)������� ����:�UR(�E��;���C�5�;�D�S��)��7!��(�E��=D�;����(�C��F�+����D��)�=D��)���One��m��rust�also�describ�S�e�what�happ�ens�at�the�cusps�but�w��re�will�gloss�o�v�er�that.�������W��Ve��ma��ry�also�describ�S�e�this�complex�analitically�are�follo�ws:���������̺0����(�pN�@�)�UR������̺0���(�N�@�)�d�so����̺0���(�pN�@�)�n�H�UR�!�����̺0���(�N�@�)�n�H�,��whic��rh�d�corresp�S�onds�to�� �h�:�UR�X���̺0���(�pN�@�)��!��X���̺0���(�N�@�).�� �Also,��conjugation�b��ry�����f\� �����d����T�p��� ��0��������-0��� �1�����+����f\�!���7pQ�maps����̺0����(�pN�@�)�to����̺0���(�N�@�)�since��,�;����l�1��f\� �����d���y�ݼp�������0�������y��0������1���������f\�!������K0��f\ �����d�����E�a����a�b��������5�pN�@�c����Ӯd�������f\�!�������Z��f\ �����d������p�������0����������0������1���������f\�!����t[��*���1�� &)�=���UR��f\� �����d�����a���)�#pb�������?�N�@�c���,\�d�����9ܺ��f\�!����� �����39����(7���K����b�{�K���������So��w��re�get����̺0����(�pN�@�)�n�H�UR�!�����f\� �����d���?��p��� %��0�������B�0��� %�1�����+���f\�!���4�Q����̺0���(�pN�@�)�������f\� �����d���ꪼp��� �W�0��������0��� �W1�����)�S��f\�!����1����*���1��>�y�n�H��!�����̺0���(�N�@�)�n�H�꨹and�this�is�� 󉍒� ؼ ����:�UR�X���̺0����(�pN�@�)��!��X���̺0���(�N�@�)�:��s����No��rw��w�e�consider�pulling�bac�k�di eren�tials�via�these�maps.�8�As�alw�a�ys,�recall�that������Y�H���V�� ��0���Z�(�X���̺0����(�N�@�)�;���� ��� ��1���)�UR=��S���̺2����(���̺0���(�pN��))���so�j�as�w��re�ha�v�e�� ��;��� ��Q�:�/�X���̺0����(�pN�@�)��!��X���̺0���(�N�@�),���w��re�j�get�� ������2���ӓ�;��� ���O���2��� �U�:�/�H���V���2�0���Z�(�X���̺0���(�N�@�)�;�� ����2�1���)�/�!��H���V���2�0���Z�(�X���̺0���(�pN�@�)�;���� ����2�1���)����and�)�th��rus�� ������2���ӓ�;��� ���O���2��� �ڹ:�t��S���̺2����(���̺0���(�N�@�))��!��S���̺2���(���̺0���(�pN�@�)).��HThese�)�are�giv��ren�b�y�� ������2���ӓ�(�f�G��)�t�=��f�qʹand,�y�if�)˼f����=���������P����*��� �;�1���̍� �;�n�=1��� �a���̽n���P�q��n9���2�n����,��then�� ���O���2���jS�(�f�G��)�UR=��p���������P����*��� �9�1���̍� �9�n�=1��� �a���̽n���q��n9���2�pn�� ]�.���ڍ���Exercise��3���B��Che��ffck�35that�� ���O���2���jS�(�f�G��)�UR=��p���������P����*��� �9�1���̍� �9�n�=1��� �a���̽n���P�q��n9���2�pn����r�ather�35than�� ���O���2���jS�(�f�G��)�UR=�������P����*��� ��1���̍� ㍽n�=1��� hW�a���̽n���P�q��n9���2�pn�� ]��.������Exercise��4���B��Show�'�that��H���V���2�0���Z�(�X���̺0����(�N�@�)�;���� ����2�1���)��!��H���V���2�0���(�X���̺0����(�N�@�)�;���� ����2�1���)�'��given�by�� ������� ���_�� ������2��� �/�is�the�same�as����that�35given�by�� �������j������ ���O���2���jS�,�which�b��ffoth�e�qual��T���̽p���]�.�fiOne�must�show�that�they�b�oth�give�� �N�����T���̽p�� ��:���������1�� ��� ԟ����X��� ҁ��UR�n�=1���Z �a���̽n���P�q��n9�� ��n�� kۿ7!���������1�� ��� ԟ����X��� ҁ��UR�n�=1����a���̽pn�� ﭼq��n9�� ��n���1�+����p���������1�� ���ɀ�����X��� ҁ�����n�=1���ȼa���̽n���q��n9�� ��pn��� b�����Another��w��ra�y�to�lo�S�ok�at�this�is�as�follo�ws:������Let�꨼'�UR�:��X�Fտ!��Y��p�.�8�Then�let��=�the�graph�of��'����X��+�����Y��p�.�8�So�w��re�get��#b���񿛍���� ����,}�,���!����p �X��+�����Y������������ ��g ��:M�!������:O�X��������M ����#������� X�Y�������as��corresp�S�ondences.�8�W��Ve�ha��rv�e�꨼ �h�:�UR��!��X��+�and�� ����:���!��Y��p�.������������������ ��g ��UR�!�������X��_�has���degree��d�UR���1.�6�So��ܼ ��O�(� ������2��1�� p �(�x�))��2���Div���h(�Y��p�),��and�deg(� ��O�(� ������2��1�� p �(�x�)))�=��d�.�6�So�w��re�ha�v�e,���for��eac��rh��n�,��Div��������x�n�� -�(�X��)�UR�!���Div����h���x�dn�� ��(�Y��p�).�8�F��Vor�꨼n��=�0,�w��re�get��Div��������x�0��D¹(�X��)��!���Div����h���x�0���l�(�Y��p�).������Applying��this�to�our�situation,�w��re�ha�v�e��T���̽p���%�H����#*�����>�X���̺0����(�pN�@�)�������K���<\� �������y�!��������{�X���̺0����(�N�@�)��������Hμ �h�#���������X���̺0����(�N�@�)�����$ŗ�And��w��re�get���B����(�E��;���C�ܞ�)��������o޽���-:����g	��UR�7!�������1������X����������D�<r��E�rغ[�p�]���0*��(�E�;�C�5�;�D�S��)�����K���5������UR�7!�������1������X����������D�<r��E�rغ[�p�]���0*��(�E�=D�;����(�C��F�+����D��)�=D��)��"o����What��happ�S�ens�on�Jacobians?������Giv��ren�35a�curv�e��X��,�EXw�e�get��J�rac�(�X��)���=��J��(�X��)�35whic��rh�is�an�ab�S�elian�v��X�ariet�y��V.��If�gen�us(�X��)���=��g�n9�,���then���dim���(�J�r�(�X��))�UR=��g�n9�.������Ov��rer��3�C�,���J�r�(�X��)�UR=��H���V���2�0���Z�(�X�Jg;����
����2�1����)����2����=H���̺1���(�X�;����Z�).�d�J�r�(�X��)��3is�co��rv��X�arian�tly��3asso�S�ciated�to��X��.�dThis�is�the���Albanese��construction�of��J�r�(�X��).������There��is�another�construction�of��J�r�(�X��)�whic��rh�is�con�tra�v��X�arian�t.������If�lQ�X�#��=�2�Y��p�,���y��rou�need�to�b�S�e�careful�ab�out�whic��rh�construction�y�ou�use�so�that�corresp�S�on-���dences��from��X��+�to��X��giv��re�the�righ�t�morphisms��J�r�(�X��)�to��J��(�X��).�������40����)EV��K����b�{�K������{��F��aGebruary�z�26,�28�and�Marc��u�h�1,�1996��b#��Scrib�S�e:�8�Sh��ruzo��T��Vak��X�ahashi,��<[email protected]>�����The��topics�co��rv�ered��in�this�w��reek's�lectures�are�������1.�Generalit��y��on�Corresp�ondences������2.�Hec��k�e��Op�erators�on��J���̺0����(�N�@�)�,��H���V���2�0���Z�(�X���̺0���(�N�@�)�;����
����2�1���)�,�and���Cot��n,(�J���̺0���(�N�@�))�������3.�Reduction��of��J���̺0����(�N�@�)��mo�S�d��D�p�������4.�The��Eic��hler-Shim�ura�Relation������5.�Galois��Action�on�T���ate�Mo�dule������6.�Shim��ura��Construction�of�an�Ab�elian�V���ariet�y�Asso�ciated�to�a�New�F���orm������7.�Galois��Represen��tations�Asso�ciated�to�New�F���orms�of�W�eigh��t�2�����1.�32Generalit���y�ffon�Corresp�s3ondences������Before��3considering�Hec��rk�e��3op�S�erators�on��J���̺0����(�N�@�),�WU�H���V���2�0���Z�(�X���̺0���(�N��)�;����
����2�1���),�WUand���3Cot���_(�J���̺0���(�N��)),�WUlet's���consider��corresp�S�ondences�in�a�general�setting.�8�Consider�a�corresp�ondence��/w������������������f�3 �/������f�3 ������������ܸ���������?������%����������������+N���جS��������S������3 S�����?������}�������������3 �w��������ּX���	��Y������,�D����This��induces�naturally�a�map�����VC��������j���������O���������:�UR�H���V����0���Z�(�Y��;����
�����1����)��!��H���V����0���(�X�Jg;����
�����1����)�:����Since�꨼X�Fտ!�UR�J�r�(�X��)�induces��Cot����(�J��(�X��))�UR�'��H���V���2�0���Z�(�X�Jg;����
����2�1����),��the�ab�S�o��rv�e��map�induces������	~Cot�����(�J�r�(�Y��p�))�UR�!���Cot��C~(�J��(�X��))�:����Also,��the�corresp�S�ondence�induces�����C���������j���������������(�:�UR�J�r�(�X��)��!��J��(�Y��p�)���b��ry��the�Albanese�functorialit�y�of��J�r�(�:�).�M�Moreo�v�er�,�3Aif�w�e�ha�v�e�� ����:�ƼJ�r�(�X��)��!��J��(�Y��p�),�3Athen��w��re���ha��rv�e��޼ ��n9���2�_��
Q�:�hh�J�r�(�Y��p�)����2�_��	��!��J��(�X��)����2�_��*��.�Z�Ho��rw�ev�er���since�w��re�ha�v�e�the�canonical�dualit�y��J�r�(�X��)�����P���hh����԰����P�=�����&'�J��(�X��)����2�_��*��,���� ����induces�꨼J�r�(�Y��p�)�UR�!��J��(�X��).�8�W��Ve��ha��rv�e�(��������j������������2���ӓ�)����2�_��	��=�UR��������������O���2���jS�.������2.�32�T��(��p��
�f�on�ff�J��(��0����(�N���)�,��H��͟��=�0�����(�X��(��0���(�N���)�;�fd�
����=�1���)�,�and���Cot���(�J��(��0���(�N���))������Assume�꨼p�UR�6�j�N�@�.�8�Consider�a�corresp�S�ondence���-���������X���̺0����(�pN�@�)������ΚU�3 �/����ΚU�3 ���Հ��������g���������?����χ�����������������=���جS���������S������3 S�����?������l�������������3 �w���������y��X���̺0����(�N�@�)�����żX���̺0����(�N�@�)�����������41����*VѠ�K����b�{�K������{��where�Vռ�jd�and���$�are�degeneracy�maps�(Consider�a�p�S�oin��rt�of��X���̺0����(�pN�@�)�as�(�E��;����Z�=p�Z�),�q�regarding����a�H1group�of�order��N���as�a�part�of��E���.).�Q|This�corresp�S�ondence�is�b��ry�de nition��T���̽p���]�.��T���̽p�� ��naturally���induces����[email protected]μT��� ���������ڍ�p��� � �=�UR� �������j������ ���O�� �������:��H���V�� ��0���Z�(�X���̺0����(�N�@�)�;���� ��� ��1���)�UR�!��H���V�� ��0���(�X���̺0����(�N�@�)�;���� ��� ��1���)�:����Iden��rtifying�꨼H���V���2�0���Z�(�X���̺0����(�N�@�)�;���� ����2�1���)�and��S���̺2���(���̺0���(�N�@�)),�this��T����2�������RA��p��� Lr�is�the�same�as�the�usual��T���̽p���]�.��������P\������H���V���2�0���Z�(�X���̺0����(�N�@�)�;���� ����2�1���)����bD�'������S���̺2����(���̺0���(�N�@�))��������g�T����2�������RA��p��� � �#�����#�UR�T���̽p�����������H���V���2�0���Z�(�X���̺0����(�N�@�)�;���� ����2�1���)����bD�'������S���̺2����(���̺0���(�N�@�))�����#V�No��rw�w/consider�the�Jacobian�of��X���̺0����(�N�@�)�=��J���̺0���(�N�@�)�whic��rh�is�an�ab�S�elian�v��X�ariet�y�o�v�er��Q�.� �v� � ѹ:����X���̺0����(�pN�@�)�^��!��X���̺0���(�N�@�)��Kinduces�� ������2��� 2%�:�^��J���̺0���(�N�@�)��!��J���̺0���(�pN�@�)��Kb��ry�the�Picard�functorialit�y�and�� ������� ��:����J���̺0����(�pN�@�)�UR�!��J���̺0���(�N�@�)��b��ry�the�Albanese�functorialit�y��V.�������Theorem��10���L���The�35action�of�� �������j������ ������2��� ��on��J���̺0����(�N�@�)��is�multiplic��ffation�by��p��+�1�.������Theorem��11���L���We�,�have�� �������\?���;� ������2���(�=�UR� ���������� ���O���2���jS�.�d?L��ffet�,��T���̽p�� ��=�UR� ���������� ������2���ӓ�.�d?Then�,��T���̽p����is�an�endmorphism�of�����J���̺0����(�N�@�)�.�fi(�T���̽p�����is�35c��ffal���le�d�����̽p�����in�Shimur�a's�b�o�ok.)�������̽p����induces�����;����� ����s����ڍ�p����ɹ:��URCot��C~(�J���̺0����(�N�@�))�UR�!���Cot��(�J���̺0����(�N�@�))���Next,��consider�����:�UR�X���̺0����(�N�@�)��!��J���̺0���(�N�@�),��mapping��x��to�the�class�of�(�x�)������(�1�).�8��>6�induces������fۼ��S��� ����h�:��URCot��C~(�J���̺0����(�N�@�))�UR�'��H���V�� ��0���Z�(�X���̺0���(�N�@�)�;���� ��� ��1���)���Iden��rtifying���Cot����(�J���̺0����(�N�@�))��and��H���V���2�0���Z�(�X���̺0���(�N�@�)�;���� ����2�1���),������2���s���RA��p��� ? �is�the�same�as��T����2�������RA��p���aʹ.��*[�����P\�������Cot����(�J���̺0����(�N�@�))����W�'����l��H���V���2�0���Z�(�X���̺0����(�N�@�)�;���� ����2�1���)���������ܼ����2���s���RA��p����ɿ#��� ��#�UR�T����2�������RA��p�������������Cot����(�J���̺0����(�N�@�))����W�'����l��H���V���2�0���Z�(�X���̺0����(�N�@�)�;���� ����2�1���)�����(�����So��+far�w��re�ha�v�e�considered�the�case��p�UR�6�j�N�@�.�-aIf��+�p�j�N��,��w�e��+need�to�c�ho�S�ose�b�et��rw�een��+� �������$:��d6�������2������and������������ڿ�ּ���O���2���jS�.��BThe���usual�c��rhoice������2���s���RA��p����E�induces�the�usual��T���̽p��f+�on��S���̺2����(���̺0���(�N�@�)).�No��rw���w�e�ha�v�e�����̽p����2��UR�End��b(�J���̺0����(�N�@�))���for��ev��rery��p�.�8��T�UR�=��Z�[�����������T���̽p����[��������Y�]�����End��b(�J���̺0����(�N�@�))��is�the�same�ring�as��T�����Z��	�}���UR�End��(�S���̺2����(���̺0���(�N�@�))).����3.�32Reduction�ffof��J��(��0����(�N���)��mo�dCd��"��p������Igusa��sho��rw�ed�that�for�all��p����6�j�N�@�,���J���̺0����(�N��)�o��rv�er��Q������Z�[1�=��X�N�@�]�has�a�go�S�o�d�reduction�mo�d��p����considering�꨼X���̺0����(�N�@�)�o��rv�er���Z�[1�=��X�N��].������On��dthe�other�hand,�ݥ�X���̺0����(�pN�@�)�=�F���̽p�����is�obtained�b��ry�attac�hing�the�t�w�o�copies�of��X���̺0����(�N�@�)�=�F���̽p�����at���their�ɠsup�S�ersingular�p�oin��rts�(those�whic�h�arise�from�sup�S�ersingular�elliptic�curv�es�o�v�er������fe|r�	n��F����
F��̽p��
o�)�a���sup�S�ersingular��?p�oin��rt��x��on�the�rst�cop�y�b�S�eing�iden�tied�with�its�F��Vrob�S�enius�transform��x����2�(�p�)��G�on���the��second.�������42����+b���K����b�{�K������{��4.�32The�ffEic���hler-Shim�ura�Relation�������Supp�S�ose�꨼p�UR�6�j�N�@�.�8�J���̺0����(�N��)�has�a�go�S�o�d�reduction�mo�d��p�.�8�W��Ve�ha��rv�e������Ob�T���̽p����2��UR�End��b(�J���̺0����(�N�@�))�UR�,���!���End��(�J���̺0����(�N�@�)�=�F���̽p���]�)�:����F��Vrob�S�enius��morphism��F���:�UR�X���̺0����(�N�@�)�=�F���̽p����!��X���̺0���(�N�@�)�=�F���̽p����of��degree��p��induces�������frob���:�UR�J���̺0����(�N�@�)�=�F���̽p����!��J���̺0���(�N�@�)�=�F���̽p�����b��ry��co�v��X�arian�t�functorialit�y��V,�and�������jv��rer����\:�UR�J���̺0����(�N�@�)�=�F���̽p����!��J���̺0���(�N�@�)�=�F���̽p�����b��ry��con�tra�v��X�arian�t�functorialit�y��V.�8�Since�the�degree�of��F��n�is��p�,�w�e�ha�v�e��ۊ����Theorem��12����L�޹v��rer��^�&������frob����=��URfrob��������v��rer����=�UR�p������The��Eic��rhler-Shim�ura�relation�is�������Theorem��13���L�޼T���̽p����=��URfrob����+����v��rer��UH�.����Note.�8�If��w��re�use��X���̺1����(�N�@�),�then�w�e�ha�v�e��T���̽p����=��URfrob����+����h�p�i��v�er����.����Idea��of�Pro�of.���First,���b��ry��the�w�ork�of�Deligne-Rapp�S�ort,���the�reduction�of��X���̺0����(�pN�@�)��mo�d���D�p��can���b�S�e�
�considered�as�t��rw�o�
�copies�of��X���̺0����(�N�@�)�=�F���̽p���]�.��*Consider�p�oin��rts�on��X���̺0����(�pN�@�)�as�isogenies�(�E�	i�!�UR�E������2�0���P�).���One��cop��ry�of��X���̺0����(�N�@�)�=�F���̽p����is�obtained�b�y�the�map������p�r���:�UR�E�	i�7!��(�E����������ںfrob��g	�����������	p!�����	r�E�������(�p�)���)�:�����The��other�cop��ry�is�obtained�b�y�the�map�����%��s�UR�:��E�	i�7!��(�E�������(�p�)����������v�Îer��g	����������mE!�����mG�E���)�:����Through��these�maps,�w��re�ha�v�e��㨍�7D�X���̺0����(�pN�@�)�=�F���̽p������P��������԰���5��=����������ō�f�N�X���̺0���(�N��)�=�F���̽p����[�������9"�X���̺0���(�N��)�=�F���̽p������Qm�fey�����꨹iden��rtifying�osup�S�ersingular�p�oin��rts�of���LO�X���̺0���(�N��)�=�F���̽p�����via����frob�������!\�Consider�꨼�h�:�UR(�E�	i�!��E������2�0���P�)��7!��E����and������:�(�E�	i�!��E������2�0���P�)��7!��E������2�0���.�8�Then��w��re�ha�v�e��Go������ ����:
�X���̺0����(�N�@�)��� 9��X���̺0����(�N�@�)�������� ̟��Q����ğ�*�Q��� ��UXQ����ട�Q���� ����Q����������
����*&�r������ �����s�������������+�������������������	 ��UX������*�����|������������
����	�s�����������[email protected]������-��󪬉��fe�������X���̺0����(�pN�@�)���,Q�id�����1-B�󪬉��fe���������Կ#������ ̟���+����� ̟������ğ���� ��UX����ട�*����� ��������������<w��������������������Q������*�Q���	 ��UXQ������Q����|���Q�����������	����������|����s�����.t�#�������:
�X���̺0����(�N�@�)��� 9��X���̺0����(�N�@�)�����������43����,sՠ�K����b�{�K������{��where���������r���=�UR�id��=���@C���s��and��������s�UR�=��F���=���@C���r�S��.�5Th��rus��the�corresp�ondence��T���̽p����can�b�e�dened����b��ry��t�w�o�corresp�S�ondences�as�follo�ws:��/�D��������I�g�X���̺0����(�pN�@�)���ٹݼX���̺0����(�N�@�)���f�}�X���̺0����(�N�@�)������A���3 �/����A���3 ���H�������Ogk���������?����B�t���������������dڝ���جS���k�����S���r�S�3 S�����?����q�̼���������r�S�3 �w��������=�����ΚU�3 �/����ΚU�3 ���Հ��������g���������?��������id����������������=���جS���������S������3 S�����?������l�F������������3 �w�����,ΰ�+�����[���3 �/����[���3 ���b�P������if����������?����Zֆ�F��������������~�ݟ��جS�����8���S�������3 S�����?�������id�������������3 �w��������)zU�X���̺0����(�N�@�)���o�%�X���̺0����(�N�@�)����y��X���̺0����(�N�@�)�����żX���̺0����(�N�@�)���Cy��X���̺0����(�N�@�)�����e�X���̺0����(�N�@�)������0�The��rst�diagram�of�the�righ��rt�hand�side�is�the�graph�of�the�F��Vrob�S�enius��F�h�:�o��X���̺0����(�N�@�)��!��X���̺0���(�N�@�)���and��the�second�one�is�its�dual.�8�F��Vrom�this,�one�can�deduce�the�Eic��rhler-Shim�ura��relation.�����5.�32Galois�ffAction�on�T���fate�Mo�s3dule������Lo�S�ok�ϟat��T���ate���̽l��!ȹ(�J���̺0����(�N�@�))�UR=����lim�����F� ������T���������J���̺0���(�N�@�)[�l��C����2���w!�]�sp�
�����Z��i?�l�����Q���̽l���g�(�l���6�=��p�).�/�Let's�ϟcall�this��X��.�This�is�the�same��UT�in��c��rhar��p��as�in�c�har�0.�������Theorem��14���L���We�35have�op��ffer�ators�35�T���̽p���]�,���frob����'��̽p�����,�and���v��rer����՟�̽p���g�on��X�$��satisfying�the�fol���lowing:������(a)�35�T���̽p�� ��=��URfrob����D��̽p�� WI�+����v��rer���UH��̽p�� ��,������(b)��35�frob����'��̽p��� ���v��rer���-5$��̽p��5Qӹ=�UR�p�,������(c)�35�T���̽p����=��URfrob����D��̽p��WI�+����p���frob������̽p����WO���W��1��$���,������(d)��35�frob�����W���'�2��" ���'�p��� 5,�����T���̽p����]�frob���WO��̽p�� �T�+��p�UR�=�0�.���Mor��ffe�over�35�X�$��has�actions�of������(1)�35�G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)�,������(2)��35�End���?����Q��!j�J���̺0����(�N�@�)����
�����Z��	'��Q���̽l��!��,������(3)��35�End���?����F���p���$A5�J���̺0����(�N�@�)���� �����Z�� '��Q���̽l��!��,������(4)�35�G��.�al�C��(�����fe Cϟ n��F���̽p���� Cϼ=�F���̽p���]�)�.���(1)��Nand�(2)�c��ffommute,��|and�(3)�and�(4)�c�ommute.�I�(4)�is�a�sp�e�cial�c�ase�of�(1)�b�e�c�ause�we�have���a�35map��D���̽p�� ���UR�G��.�al�C��(�����fe Cϟ n��F���̽p���� Cϼ=�F���̽p���]�)��wher��ffe��D���̽p����URG��.�al�C��(�����fe #�� n��Q��� #��=�Q�)��is�a�de��ffc�omp�osition�35gr�oup.����P��oin�t.� ���There�׌is�a�F��Vrob�S�enius�elemen��rt��'���̽p�� c��2��GG��.�al�C��(�����fe Cϟ n��F���̽p���� Cϼ=�F���̽p���]�)�and�a�F�rob�S�enius�endmorphism����frob������̽p�� ���2��UR�End���b����F���p���#cR�J���̺0����(�N�@�)���� �����Z�� '��Q���̽l��!ȹ.�8�On��T��Vate�mo�S�dule��X��,��frob���z���̽p��!,��acts�as��'���̽p���]�,�and�w��re�ha�v�e������5��'��� ���2���ڍ�p���r�����T���̽p���]�'���̽p���+��p�UR�=�0���whic��rh��reminds�Ca�yley-Hamilton�theorem�with��det��<d�'���̽p�� ��=�UR�p��and��tr�� 0�'���̽p���=�UR�T���̽p���]�.�8�Also,�w��re�ha�v�e�����+E�T���̽p�� ��=�UR�'���̽p��r�+����p�'���̽p���� yӟ� ���1���O�:����6.� �Shim���ura�K�Construction�of�an�Ab�s3elian�V���fariet�y�Asso�s3ciated�to�a����New�ffF���form������Consider���T���̺0��V�=�UR�Z�[�����������T���̽n����N��������L�]�where�(�n;���N�@�)�=�1.�8�W��Ve�ha��rv�e�������T�T���̺0��j�� ����Q�UR�'���������Y��� 8獑=R�f��� n�E���̽f���� �����44����-��K����b�{�K������{��where��f�[�runs�o��rv�er�the�set�of�new�forms�of�lev��rel�dividing��N�S��mo�S�dulo�conjugation�b�y��G��.�al�C��(�����fe #�� n��Q��� #��=�Q�).����Also,��w��re�ha�v�e������5�T���� ��Q�UR�'���������Y��� 8獑=R�f��� n�A���̽f��� �t��where�꨿A���̽f�� aǹis�an��E���̽f���algebra.�������Exercise��5���B��Show��35�dim����}��̽E��i?�f���%MֿA���̽f���q�=�UR#��35of� mdivisors�of������Fu��S\ݽN��M��콉fe,۟�'�N��"�(�f��Ǻ)�����f&T�wher��ffe�35�N�@�(�f�G��)��is�the�level�of��f��.�� #Z����Example��4���DUQ�Supp��ffose�35�N��6�=�UR�N�@�(�f�G��)�q��n�wher�e��q�Ë�6�URj�N�@�(�f�G��)�.�fiThen�we�have�������3�A���̽f���q�=�UR�E���̽f��w �[�U�@�]�=�(�U���� ��2���������a���̽q�����U�댹+��q�n9�)�:������Exercise��6���B��Write�35�A���̽f�� �T�in�gener��ffal�as�������E���̽f��w �[�����������U���̽q���������������]�=�(�relations��+� )����wher��ffe�35�q�n9�j�����Fu�����N��33�콉fe,۟�'�N��"�(�f��Ǻ)������A�.����More��explanation�ab�out�the�ab�o��v�e��example.�����A���̽f�� u/�is��the�algebra�generated�b��ry�the��T���̽p���m�acting�on�the��C��v�ector�space�generated�b�y�the�forms��[��f�G��(�d �W �)��where��d�j�����Fu�����N��33�콉fe,۟�'�N��"�(�f��Ǻ)������A�.�8�T���̽p����acts�as�the�scalar��a���̽p���if��p�UR�6�j�N�@�.�8�If�꨼q�n9�j�N��,��T���̽q���P�=�UR�U���̽q����acts�as�����f\� �����d�����a���̽q����)Α�1��������T��q���)Α�0�����4����f\�!���<�9�.�� ��An��ab�elian�sub��v��@ariet�y���A���̽f�� � �of��J���̺0����(�N�@�).���Let�K�����̽f���ɹb�S�e�the�pro�� jection�of��T���̺0��%� �e��Q�UR�'�������Q��� ����f��>8�E���̽f���o��rv�er�K��E���̽f��w �.��Then�de ne��A���̽f���q�=�UR����̽f���(�J���̺0����(�N�@�))�(����̽f���ɹis�not���exactly��qan�endmorphism�but�is�in��End���0(�J���̺0����(�N�@�))�l� ��Q�).�.�In��qthe�category�of�ab�S�elian�v��X�arieties�up���to�һisogen��ry��V,��ׄ�����Q��IK�A���̽f�� Iڹacts�on��J���̺0����(�N�@�)�and�pulls�apart��J���̺0���(�N�@�)�in��rto�������Q��� D����f�����A���̽f��w �.�0�W��Ve�ha�v�e�������P�������f����A���̽f���q�=�UR�J���̺0����(�N�@�)���and��ꨟ����Q��� \q���f��ӎ�A���̽f���q�!�UR�J���̺0����(�N�@�)��is�an�isogen��ry��V.������Also,�꨼J���̺0����(�N�@�)�=�C��can�b�S�e�considered�as����y��J���̺0����(�N�@�)�UR=��Hom����(�S���̺2���(���̺0���(�N�@�))�;����C�)�=H���̺1���(�X���̺0���(�N��)�;����Z�)���(call�6�it��V�N8=L�).���Then��V��¹=��UR�����L��qļV���̽f�����and��A���̽f���q�=�UR�V���̽f��w �=�(�V���̽f����\�:¼L�)�where�the�basis�of��V���̽f���consists�of��f�G��(�d �W �)'s���as��w��rell�as�conjugates�of�them.����Sp�ecial��case�when�the�lev��el�of��f����is��N�@��.����No��rw,� �let's��yconsider�a�sp�S�ecial�case�where��f�x�is�a�new�form�(�N�@�(�f�G��)�UR=��N��).��{Then��yw��re�ha�v�e��A���̽f���q�=�UR�E���̽f��w �.���Also����d��0�=��dim��6v�A���̽f�� O�=��dim���6x����Q��"�A���̽f���=�[�E���̽f���:��Q�].���Fix����f����and�let��A��0�=��A���̽f��$��and����E�UG�=��E���̽f��w�.�Let����T���ate���̽l�����b�S�e��\the��Q���̽l��!ȹ-adic�T��Vate�mo�dule�of��A���̽f������P���
]�����԰���
v��=������~�Q����2��2�d��RA�l���	��.���Action�of��E��9�
�����Q��'��Q���̽l���$�on��T���ate���̽l���is�������Q���H%����j�l�����E���̽��uZ�.���Th��rus����T���ate���̽l��w�=��UR�����Q��� �����j�l��:��T�ate���̽�� �where�꨼T�ate���̽���is��an��E���̽���v��rector�space.�8�W��Ve�ha�v�e�������Lemma��3����<uQ�dim���P ���̽E��i?����]��T���ate���̽��ʬ�=�UR2�35�for�al���l���j�l�C��.�fiThat�is,��T�ate���̽l��T��is�fr��ffe�e�35of�r��ffank�2�over��E�^�� ����Q���̽l��!��.����Pr���oof.�湼A���̽f���q�=�UR�V���̽f��w �=L���̽f����where�SƼL���̽f���=��V���̽f��홿\�vz�L�.���L���̽f��� ��Q�Sƹa ords�an�action�of��E���̽f����(a� eld).��So��L���̽f��홿 ��Q�����is��a�v��rector�space�o�v�er��E���̽f�� aǹof��dim���B=�UR2.�8�On�the�other�hand,��T���ate���̽l�� p�is�(�L���̽f��!ǿ ����Q�)�� �����Q�� � �Q���̽l��!ȹ.��� ����45����.�{��K����b�{�K������{��7.�32Galois�ffRepresen���tations�Asso�s3ciated�to�New�F���forms�of�W�eigh���t�2�������G��.�al�C��(�����fe #�� n��Q��� #��=�Q�)��acts�on��T���ate���̽l�� p�whic��rh�is�compatible�with��E�^�� ����Q���̽l��!ȹ.�8�That�is,�w�e�ha�v�e���|��OF��G��.�al�C��(�����fe #�� n��Q��� #��=�Q�)������K��^f���i?�l������!����� �Aut���̽E�r�� �Q��i?�l���Z�T���ate���̽l��w�=��URGL����(2�;���E�^�� ����Q���̽l��!ȹ)�UR=���������Y��� 8獑>4����� n�GL��"� (2�;�E���̽��uZ�)�:��"�;��Then��w��re�ha�v�e�����O����̽l��w�=����UR�����M����qż���̽�����where������y����̽��ʬ�:�UR�G��.�al�C��(�����fe #�� n��Q��� #��=�Q�)��!��Aut���̽E��i?���� �Y�T���ate���̽�����If��O�p�UR�6�j�l�C�N�@�,���then�����̽�� D��is�unrami ed�at��p��and�����̽��uZ�(�'���̽p���]�)�has�a�w��rell�de ned�(trace,�det)�c��rharacteristic���p�S�olynomial��(�X��)�giv��ren�as�������Theorem��15���������(�X��)�UR=��X���� ��2��\/�����a���̽p���]�X��+�+��p����wher��ffe�35�a���̽p�����is�the�image�of��T���̽p���in��E���̽f��w �.�����Idea��of�Pro�of.�8�By��Ca��ryley-Hamilton,�w�e�ha�v�e������(1)�꨼'����2��2��RA��p���r�����(�tr�� #�����̽��uZ�(�'���̽p���]�))�'���̽p���+��det���d����̽��uZ�(�'���̽p���]�)�UR=�0���By��Eic��rhler-Shim�ura,�w�e�ha�v�e������(2)�꨼'����2��2��RA��p���r�����a���̽p���]�'���̽p���+��p�UR�=�0���Th��rus���it�suces�to�pro�v�e�(�det��Q�����̽��uZ�)(�'���̽p���]�)���=��p����whic�h�is�equiv��X�alen�t�to��det�� Z����̽�� *�=���� ���̽l���f�b�y�Ceb�S�otarev���densit��ry��_theorem�where�� ���̽l��J��:�(�G��.�al�C��(�����fe #�� n��Q��� #��=�Q�)��!��Z����2����RA��l��� ����E����2������RA����� r��is��_the��l��-adic�cyclotomic�c��rharacter.�qA���complete��pro�S�of�of�this�theorem�will�b�e�giv��ren�next�w�eek.��(V��Marc��u�h�z�4,�1996��b#��Scrib�S�e:�8�J�� anos��Csirik,��<[email protected]>�������Last�)�time�w��re�had�some�w�eigh�t�2�eigenform��f�qʹon����̺0����(�N�@�)�of�exact�lev�el��N�@�.��H(���̺1����(�N��)�)�w�ould���giv��re��an�in�teresting�v��X�arian�t�whic�h�w�e�shall�consider�brie y�at�the�end�of�the�lecture.)�2�W��Ve���found�}some�ab�S�elian�sub��rv��X�ariet�y�}�A���̽f�� �B��:#�J���̺0����(�N�@�),�N�it�has�an�action�on�it�b��ry��T�l�� ��Q�}�whic�h�acts���through��the�quotien��rt��E���̽f�� ��=��h�Q�(��:���:�:��ʞa���̽n���P�(�f�G��)�����:�:�:���),�iand��[�E���̽f���:��h�Q�]�=��dim�� ~��A���̽f��w �.��E���̽f�� O.�is��a�totally�real���algebraic��n��rum�b�S�er� eld.������W��Ve��7ha��rv�e�disco�v�ered�that��T��Va���>���̽��7۹(�A���̽f��w �)�is�free�of�rank�2�o�v�er�the���-adic�completion��E���̽f��R� ���3�Q���̽�����=���������Q��� qɟ���j�l���]�E���̽��uZ�,��and�the�T��Vate�mo�S�dule�breaks�apart�th��rus:�� ������\T��Va����=��̽��܋R�=����UR�����Y��� 8獑>4����� n�T��Va��� ~,��̽��$�:��"�;����W��Ve��also�ha��rv�e�����#����̽��ʬ�:�UR�G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)��!���Aut���=��̽E��i?�����"�p�T��Va���0D.��̽��9ڹ=��GL����(2�;���E���̽��uZ�)�:��������46����/�f��K����b�{�K������{��F��Vor�꨼p��UV�6�URj����N�@�,�����̽��uZ�(�frob������̽p��WO�)�satises������/(�X������2��\/�����a���̽p���]�X��+�+��p:��Z���(Note�ۓthat��a���̽p��	��is�w��rell-dened�as��E���̽f��
f��,���!��d�C��via��f�G��:���a���̽p���is�a�certain�eigen��rv��X�alue��2��d�C�.)��By�the���Ca��ryley-Hamilton��theorem,�����̽��uZ�(�frob������̽p��WO�)�also�satises������b̼X������2��\/������tr��
�.����̽��uZ�(�frob������̽p��WO�)���+��det���b����̽���(�frob������̽p��WO�)�UR=�0�:���썍��Theorem��16���L���(1)��35�tr��V�����̽��uZ�(�frob������̽p��WO�)�UR=��a���̽p���]�.������(2)��35�det�������̽��uZ�(�frob������̽p��WO�)�UR=��p�.������Note���that�(2)�implies�(1)�b��ry�the�Ca�yley-Hamilton�theorem:��suse�that�����̽��uZ�(�frob������̽p��WO�)�is�an���in��rv�ertible��matrix�and�subtract�the�ab�S�o��rv�e��t�w�o�p�S�olynomials�that�it�is�kno�wn�to�satisfy��V.������T��Vo��pro��rv�e�(2),�w�e�need�the�follo�wing�\true�theorem":������Theorem��17���L���Using�35����̽��,�to�denote�the���-adic�cyclotomic�char��ffacter,��������ڹdet��������̽��ʬ�=�UR����̽����:������Once��this�theorem�is�pro��rv�ed,��w�e�can�substitute��frob���z���̽p��!,��(on�b�S�oth�sides)�and�get�(2)�ab�o��rv�e.���v����Before�CUtac��rkling�the�true�theorem,���w�e�need�to�talk�ab�S�out�a�generalization�of�the�W��Veil���pairing��that�w��re'll�need�in�the�pro�S�of.������W��Ve��ha��rv�e�an�isogen�y�����1��A���̽f���q�,���!�UR�J���̺0����(�N�@�)��������������g	��!������J���̺0���(�N��)�����_��	��!�UR�A������_���ڍ�f���*��:��Z���This��is�a�p�S�olarisation.������Generally��V,���for��sa�p�S�olarisation�����̽L��
!K�of�an�ab�elian�v��X�ariet��ry��A��dened�b�y�the�line�bundle��L�,���w�e���need��������{6����̽L��	�*�:�UR�A������te�!�������tg�Pic�����b���x�0���d�A�UR�=��A�����_����������Id�a�����te�7!������tg�T������������ڍ�a���aʼL����
��L������1��\|�;������where�?��T���̽a��CZ�stands�for�translation�b��ry��a�.���By�the�theorem�of�the�square,�a�this�is�a�homomorphism.����W��Ve��also�require�that�����̽L��
2��b�S�e�an�isogen��ry�(�L��needs�to�b�e�ample).������The��sW��Veil�pairing�has�a�natural�generalization�to��A�UR�=��A���̽f��w�:�	�F�or��seac��rh��n�,��Jit�is�a�natural�map������j�A�[�n�]������A�����_��*��[�n�]�UR�!�����̽n�����whic��rh��is�a�bilinear�p�S�erfect�pairing.�8�Giv�en���UR�:��A��!��A����2�_��*��,��w�e�get�����j�A�[�n�]������A�[�n�]�UR�!�����̽n�����whic��rh�&is�an�alternating,�4�but�not�p�S�erfect,�pairing.���W��Ve�can�mak��re�it�p�S�erfect�b�y�setting��n��W�=������2�r������and��taking����lim��
s_�����:��(�� ������������qƍ�

U�r������<b�:��s_��o��(�:;���:�)�UR:��T��Va������̽�����A�������T��Va���f��̽��L�A�UR�!���Q���̽���qŹ(1)�=�(���lim��
s_�����:��>6� ������������qƍ���r������Q�����̽�����r���%V�)����
�����Z��i?������E�Q���̽���฼:��������47����0�S��K����b�{�K������{����In��fact�for�a�map�of�ab�S�elian�v��X�arieties��t�UR�:��A��!��B���,��w��re�ha�v�e�for������2�UR�B������2�_��
�^�and��a��2��A�,��b�����(�ta;�����O�)���̽B��
���=�UR(�a;�t�����_��*����)���̽A����:������No��rw��qtak�e��A�UR�=��B�ww�and�recall�our�isogen��ry����:��A��!��A����2�_��
!�ab�S�o��rv�e.�4#Then�taking��t�UR�2��T���̺0��M��
����Q�,��Iw��re����get������E(�ta;���a�����0���9�)�UR=�(�a;�ta�����0���9�)�:���i����T��Vo��9put�this�in��rto�con�text,��note�that�since����is�an�isogen�y��V,��it�has�an�in�v�erse�if�w�e�thro�w���in�Nwdenominators,�gkso�w��re�can�tak�e��t��5�2���End���A�Nw�in�to��t����2����9�2���5�End���A�
��Q�.�dMThis�Nwis�called�the��R��ffosati���involution�꨹asso�S�ciated�to�the�p�olarisation���.�8�It�follo��rws�easily�that�in�this�case��b����.C(�ta;���a�����0���9�)�UR=�(�a;�t���������a�����0���9�)�:������Lo�S�oking��Sat��J���̺0����(�N�@�)�and�its�natural�p�olarisation,�9=if��t��2��T��S�comes�from�a�corresp�ondence����then�꨼t����2������comes�from�its�dual.������In��our�case,���UR�=��w�=R�is��an�in��rv�olution,��so��t����2���V�=�UR�w�R�tw��.�8�W��Ve��ha��rv�e��b�������T���̺0��j��
����Q�UR���T��
��Q�����End��b(�J���̺0����(�N�@�))���and���~���2�����acts��~as�the�iden��rtit�y��~on��T���̺0��q^�
��Z�Q�.�VbIn�general,���w�R�tw�G(�is�not�alw��ra�ys��~in��T��
��Q�:�L�if��t�f�=��T���̽p���۹for�����p�j�N�@�,�꨼t����2������do�S�es�not�comm��rute�with��t�.�8�Note�that������-dim�����u����Q������T����
��Q�UR�=��dim���ꚟ���C�� ~��S���̺2����(���̺0���(�N�@�))���and������}�dim����)����Q�����T���̺0��j��
����Q�UR�=���������X�������c��d�j�N������dim���'\a����C��0�L�S���̺2����(���̺0���(�d�))�����new��Aռ:��+b�����Let's��try�to�pro��rv�e��our�determinan��rt�assertion�no�w.�8�Restrict�����H(�:;���:�)�UR:��T��Va������̽�����A�������T��Va���f��̽��L�A�UR�!���Q���̽���qŹ(1)���to������'�h�:;���:�i�UR�:��T��Va������̽��,j�A�������T��Va���f��̽�����A�UR�!���Q���̽���qŹ(1)�:���i��This�Iis�an�alternating�pairing.��Since�the�pro��jectors�come�from�elemen��rts�of��E���,�iZwhic�h�are�their���o��rwn��in�v�erses�with�resp�S�ect�to�the�pairing,�this�pairing�is�also�non-degenerate.������The�&bpairing�in�Galois-in��rv��X�arian�t�&bso�for���)5�2���G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)�and��x;���y��2�����T��Va������̽����,�5Q�h��n9x;��y��i����=�����2���_��h�x;���y��i�,���and��since���X�acts�on���Q���̽����(1)�b��ry�m�ultiplication�b�y�����̽����(��n9�),�w�e�get������K֟�������׿h�x;���y�n9�i�UR�=�����̽����(���)�h�x;���y��i�:�������F����ancy���pr���oof.�8�h�tx;���y�n9�i�UR�=��h�x;�ty�n9�i�꨹for��t�UR�2��E���̽��uZ�,��so��h�:;�:�i��factors�through��D���=��UR�����V����*������2���̍����E��i?������P�T��Va���&����̽��,'�:��������h�:;���:�i�UR�:��D���!���Q���̽���qŹ(1)�:��������48����1����K����b�{�K������{��The��left�hand�side�is�one�dimensional�o��rv�er�꨼E���̽��uZ�.�8�So�w��re�get�����u�2�h�:;���:�i�UR2���Hom����۟���Q��i?����'Gù(�D�S�;���Q���̽���q�(1))�=��Hom����۟���Q��i?����'Gù(�D�S�;�E���̽��uZ�(1))�;����where��mthe�equalit��ry�is�through�comp�S�osition�with�the�trace.�~/So�w�e�got��D�����P�������԰�����=������-�E���̽��uZ�(1),���the�left����hand��side�the�determinan��rt�of�our�represen�tation,��|the�righ�t�hand�side�acted�on�b�y�����̽����.�aDSo�w�e���pro��rv�ed��������det��������̽��ʬ�=�UR����̽����:�������Non-f����ancy��pr���oof.�
�&�Let�������2�w�G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�),�7�and�tak��re��E���̽��uZ�-linearly�indep�S�enden�t�v�ectors����x;���y�Ë�2��UR�T��Va������̽�����.�8�Set��������Lļ�n9x��������=������9��a�(��n9�)�x����+��b�(���)�y�����������7�n9y��������=������9��c�(��n9�)�x����+��d�(���)�y�:������W��Ve��w��ran�t�to�sho�w�that����x�׼�Ȅ�=�UR��s2�(��n9�)�=��a�(���)�d�(���)������b�(���)�c�(���)�������̽����(���)�UR=�0�:������Observ��re��that���������h�ax;���dy�n9�i�UR�=��h�adx;�y�n9�i�;����and����};1�h�by�n9;���cx�i�UR�=��h�bcy�;���x�i��=��h�x;�bcy�n9�i��=��h�bcx;�y�n9�i�:����Then��since����|+1�h�bdy�n9;���y��i�UR�=��h�by�n9;�dy��i�UR�=��h�dy�n9;�by��i�UR�=��h�bdy�n9;�y��i�;����if��follo��rws�that�����e��h�by�n9;���dy��i�UR�=�0�;����and��similarly������@�h�ax;���cx�i�UR�=�0�:������Therefore�����P�#��(��n9�)�h�x;���y��i�UR�=��h��n9x;��y��i�UR�=��h�ax����+��by�n9;�cx��+��dy�n9�i�UR�=��h�(�ad������bc�)�x;�y�n9�i�;����So�.��h��s2x;���y�n9�i��(�=�0�for�an��ry�c�hoice�of�linearly�indep�S�enden�t�v�ectors��x;���y�n9�.�
So�w�e�pro�v�ed�once�again����that����۔��Ȅ�=�UR0�:��������49����2����K����b�{�K������{��The��case�of�����̺1����(�N�@�)��@�F��Vor�*ja�newform��f�G��,�:[w��re�ha�v�e��f�G��j�T���̽n��	j)�=��ټa���̽n���P�(�f��)�*jfor�all��n����1,�:[and��f�G��jh�d�i��=��"�(�d�)�f�ri�for��d��2��(�Z�=��X�N�@��Z�)����2�����.����Rep�S�eating��Lthe�construction�w��re�get�an�ab�elian�v��X�ariet��ry��A���̽f��
k�with�an�action�of�a�n�um�b�S�er�eld����E���̽f��w�.�/�Ho��rw�ev�er���in�this�case��E���̽f��	F�is�not�necessarily�real:�+but�if�it�isn't�then�it�is�equipp�S�ed�with�a���canonical��complex�conjugation:��8���&�fe�Q�]ڍ�a���̽n����a��=�UR�"�(�n�)����2��1��\|�a���̽n���P�.�8�Also�w��re�ha�v�e��"�(�d�)�UR�2��E�����������%��f���Џ�.������In��this�case,�w��re�ha�v�e��������Theorem��18���L���(1)��35�tr��V�����̽��uZ�(�frob������̽p��WO�)�UR=��a���̽p���]�.������(2)��35�det�������̽��uZ�(�frob������̽p��WO�)�UR=��"�(�p�)�p�.������This��can�b�S�e�pro��rv�ed��from�the�follo��rwing:������Theorem��19���L�޼"�Gs�c��ffan�b�e�c�onsider�e�d�r�epr�esentation�of�the�Galois�gr�oup.��Stil���l�using�����̽��@Y�to�denote���the�35��-adic�cyclotomic�char��ffacter,������	�det���Sü���̽��ʬ�=�UR�"���̽����:�������Signs.���There�7�is�less�gratuitous�self-dualit��ry�going�on�here�than�in�the�case�of����̺0����(�N�@�),�J�so���w��re��shall�ha�v�e�t�w�o�o�S�ccasions�to�mak�e�c�hoices.������F��Vor���ev��rery��p�,��s�T���̽p��_C�is�dened�as�a�corresp�S�ondence�on��X���̺1����(�N�@�),�with�the�same����u�and���B5�that�w��re���ha��rv�e��used�b�S�efore.�8�Do�w��re�w�an�t�to�do�(�T���̽p���]�)����2������or�(�T���̽p���)���������?������W��Ve�[email protected]��rv�e�another�c�hoice�in�Eic�hler-Shim�ura.�	0�What�do�S�es��X���̺1����(�N�@�)�classify?��Either��an���elliptic���curv��re��E����and�a�p�S�oin�t��P����of�order��N�5ù(whic�h�amoun�t�to�a�pair�(�E��;����Z�=n�Z�f��,���!��E��));����Or��߹a���pair���(�E��;������̽n�����,���!�UR�E��).��The�second�one�is�more�suitable�for�us�(and�for�use�with�the�T��Vate�curv��re),���and��!it�forces�us�to�use�the�Picard�functorialit��ry�of�the�Jacobians.�2JTherefore�w�e�shall�mak�e����T���̽p����:�UR�J���̺1����(�N�@�)��!��J���̺1���(�N�@�)�͕so�that�(�T����2�����_��RA��p����v�)����2������on��H���V���2�0���Z�(�X���̺1���(�N�@�)�;����
����2�1���)�UR=��S���̺2���(���̺1���(�N�@�))�͕is�the�classical��T���̽p���]�.�//(Here���regard�꨼T����2�����_��RA��p�����as�an�elemen��rt�of��End���g(�J���̺1����(�N�@�))�b�y�the�auto�S�dualit�y�of��J���̺1����(�N�@�).)��'����6�z�Marc��u�h,�1996��b#��Scrib�S�e:�8�Da��rvid��M�Jones,��<[email protected]>��<]����T��Vo�S�da��ry��w�e�will�discuss�the�pro�S�cess�of�reducing�a���-adic�represen�tation�mo�S�d���.������Recall:��7Let�<S�f���=��������P����*���"N�1���̍�"N�n�=1���"��a���̽n���P�q��n9���2�n��Rܹb�S�e�a�w��reigh�t�<S2�newform�on����̺0����(�N�@�).�	-�That�is,����f��R�has�trivial���Neb�S�en��rt�ypus�Pand�is�new�of�lev��rel��N�@�.�i9W��Ve�lo�ok��red�at�the�eld��E���=��Q�(�:::a���̽n���P�:::�).�i9F��Vor�eac�h�prime�����%�of��E��<�lying�ab�S�o��rv�e�%a�xed�rational�prime���,�let��E���̽��	��b�e�the�completion�of��E��<�at�the�v��X�aluation���corresp�S�onding��to���.�8�Then�w��re�created�a�represen�tation��I�����ۼ���̽��ʬ�:�UR�G��.�al�C��(����!�����Q���
#��=�Q�)��!���GL������̺2����(�E���̽��uZ�)���suc��rh��that��det��<d����̽��ʬ�=�UR����̽����,�the���-adic�cyclotomic�c�haracter,�and��tr��
0����̽��uZ�(�frob������̽p��WO�)�UR=��a���̽p����for�꨼p��-��N�@�.��#�����Lemma��4���<uQ����̽��	���is��e��ffquivalent�to�a�r�epr�esentation�with�values�in���GL����<��̺2���@�(�O�UV�)�,�k�wher�e��O��Ź=��the�ring����of�35inte��ffgers�in��E���̽��uZ�.��������50����3�n��K����b�{�K������{��Pr���oof.����An�)equiv��X�alen��rt�statemen�t�to�the�lemma�is:��there�is�an��O�UV�-lattice��L��in��V����(the�v�ector����space��on�whic��rh��G��.�al�C��(����!�����Q���
#��=�Q�)�acts�via�����̽��uZ�)�whic�h�is��G��.�al�C��(����!�����Q���
#��=�Q�)-stable.������Pic��rk��ian�y�lattice��L���̺0��	Km�in��V��p�.�#The�set�of�lattices��g�n9L���̺0���with��g��*�2�f�G��.�al�C��(����!�����Q���
#��=�Q�)�is�nite.�#This�is���b�S�ecause�h��G��.�al�C��(����!�����Q���
#��=�Q�)�is�compact�and��L���̺0��	(��is�discrete.���So�w��re�tak�e��L�+��=�������P����ҟ��g���μg�n9L���̺0����,��whic�h�h�is�stable���under��Galois.����������So��w��re�are�led�to�ask�the�follo�wing������Question:�#>Can��dw��re�dene������������̽�������=�UR����̽���	4��mo�S�d��!����b�y�taking�an�equiv��X�alen�t�represen�tation�as�in�the���lemma��and�reducing�it���Pmo�S�d�� Z���?������More�sp�S�ecically��V,�let��F����=��O�UV�=�,�a�nite�eld.���Then�the�map��O��U!����F��induces��GL�������̺2��b��(�O�UV�)��!�����GL�������̺2��O��(�F�).�8�Can��w��re�dene�����������̽����Z�b�y�the�comp�S�osition�������y��Ë�:�UR�G��.�al�C��(����!�����Q���
#��=�Q�)��!���GL������̺2����(�O�UV�)��!���GL������̺2���(�F�)?������The���answ��rer�is�no,���b�S�ecause�this�map�dep�ends�on�the�c��rhoice�of��L��in�the�lemma.�N�So�w�e�do����the��follo��rwing:�������Denition��11�����U4,����Raü���̽����c/A�=�H�(����:��G��.�al�C��(����!�����Q���
#��=�Q�)��!���GL����K��̺2���O�(�F�))����2�ss��4��,��Ithe�@�semisimplic��ffation�of�the�map�de-���scrib��ffe�d�35ab�ove.�fiThat�is,������1.�fiIf�35���n�is�irr��ffe�ducible,��������35����̽����
U�=�UR��n9�.��U]����2.�fiIf�35��Ë����UR��f\� �����d���?�����"r���������d�0���!�Ƽ�����.3��f\�!���6��,�then����n9���2�ss��
�1�=�UR���7�������O�.��"U^��No��rw��w�e�list�the�k�ey�prop�S�erties�of�����������̽����ov�:������(a)�������꨼���̽����Z�is��semisimple.������(b)���det��������<d����̽����#��=�UR(����̽���㎹mo�S�d�� hҼ�).������(c)���tr��������
0����̽�������(�frob������̽p��WO�)�UR=�(�a���̽p�����mo�S�d��!7I��)��2��F�;����8�p��-��N�@�.������Theorem��20���L���(Br��ffauer-Nesbitt�L�The�or�em)�Supp�ose��;�������2�0���ar�e�two�r�epr�esentations�as�ab�ove,�z�b�oth���semisimple,�35with�the�same�tr��ffac�es�35and�determinants,�then�������P���UR����԰���n:�=������������2�0���9�.������So�������꨼���̽����Z�do�S�es��not�dep�end�on�the�lattice��L�;�it�is�w��rell-dened.������No��rw�DClet's�view�this�situation�in�a�new�w�a�y:���T���̽n��I�2�����End���T(�S���̺2����(���̺0���(�N�@�)))�;����T����=��Z�[�:::T���̽n���P�:::�]�������End���(�S���̺2����(���̺0���(�N�@�)))������T��Vak��re���m�UR���T�,�a�maximal�ideal,�and��T�=�m��=��k��o�=��F���̽���������r�.������Prop�osition��5���U���Ther��ffe�35exists�a�(unique)�semisimple�r�epr�esentation�������������m��
3��:�UR�G��.�al�C��(����!�����Q���
#��=�Q�)��!���GL������̺2����(�k�g�)����such�35that:�������1.�fi������m��j�is�35unr��ffamie�d�outside��N�@��.������2.��fi�tr���������m���5�(�frob������̽p��WO�)�UR=��T���̽p���]�8�p��6�j�N�@��.������3.��fi�det���%������m���5�(�frob������̽p��WO�)�UR=��p�8�p��-��N�@��.��������51����4���K����b�{�K������{��Pr���oof.����First��of�all,��it�is�enough�to�pro��rv�e��this�for��T���̺0��F��=����Z�[�:::T���̽n���P�:::�]����T�,�where���T���̺0��ǡ�is�formed����using��only�those��T���̽n��	���with�(�n;���N�@�)�UR=�1.�8�Then���T���̺0����=�m���̺0��V�=��k���̺0�����k�g�.���(?������e����H��T���̺0��j��
����Q����SM�=����j��E���̺1��j������:::����E���̽t�����������-�����S���A��j����������S��VE�j����������T���̺0�����:e����ۏ�O���̺1��j������:::���O���̽t����(ι=�UR�O���������-�����S���A��j����,ue�����S��6ʺ�j����������m���̺0�����SM�=���/�h������2�(����W��Ve�zNcan�mak��re���L��������̽�������:����G��.�al�C��(����!�����Q���
#��=�Q�)��!���GL����-��̺2��M1�(�O�UV�=�).�	��This�zNhas�the�required�prop�S�erties�of�������m�����except��$that�it�tak��res�v��X�alues�in��GL������̺2���ù(�O�UV�=�)�UR����GL������̺2����(�k���̺0����).�_�O��=��$�b�S�eing�bigger�than��k���̺0��L(�is�a�p�oten��rtial���obstruction��to�dening�������m��
3��=���'����UR����̽����� �.������But,���the���c��rharacteristic�p�S�olynomials�of����<�������̽����P��(�g�n9�)�ha�v�e�co�S�ecien�ts�in��k���̺0����,���and�since�the�Brauer���group���of�a�nite�eld�is�trivial,���there�is�a�mo�S�del�for���g��������̽������o��rv�er����k���̺0����.��W��Ve�tak��re�������m��
sǹto�b�e�this�mo�del.����������Let��us�digress�in�order�to�explain�in�a�dieren��rt�w�a�y�what�w�e'v�e�just�done.������Recall:�C�T���̺0��n#�
���Q�]��=��E���̺1������:::����E���̽t�����and��eac��rh�factor��E���̽i��T��corresp�S�onds�to�a�newform�of�lev�el��M����for��some��M�@�j�N��.�8�Supp�S�ose�꨼E���̺1�����corresp�onds�to�a�newform�of�lev��rel��N�@�.������W��Ve� get��J���̺0����(�N�@�)��S���A���̺1�������:::����A���̽t���ʹ,�-}where� w��re�ha�v�e��dim���e�A���̺1��pW�=��S[�E���̺1���:��Q�].��FAlso,�-}�T���a���̽����A���̺1���#�is� free�of���rank��2�o��rv�er�꨼E���̺1��j��
�����Q��
��Q���̽����.������But��pif��E���̽i���J�corresp�S�onds�to�a�form�whic��rh�is�new�of�lev�el��M���;���M�@�j�N�;�M��6�6�=�UR�N�@�,��then��pthis�do�S�esn't���w��rork.�8�W��Ve��m�ust�go�bac�k�to����̺0����(�M�@�).�������Question:�<*Let�lM�A�1��=��A���̺1������J���̺0����(�N�@�).���Then�w��re�ha�v�e��T�1��!���End��>�(�A���̺1����).���Let����.�����T���<'�b�S�e�the�image�of���this��map.�8�So�����q�����T�������URO���̺1����,�but�����q�����T���8ݹma��ry�not�b�S�e�in�tegrally�closed.������But��"there�exists�a��Z���̽����-adic�T��Vate�mo�S�dule:��ӼH���̽���(�A�)���=�������Q���!r���m�����H�����m���5�(�A�)��"whic��rh�is�isomorphic�as�a���group�m�to��Z����2��2��j�dim���A��RA������.�F�H���̽����(�A�)�has�an�action�of����/������T���}&�
����Z���̽��N8�=��UR�����Q�������m�����g������N�T�����m����%��.�F��Vor��m�j��,���w��re�ha�v�e��H�����m��
L�free�of�rank���2��o��rv�er�����q�����T�����m����,j�.������Let����J�vR�=�Y�J���̺0����(�N�@�)�;����T�����End��f��J�r�.�Let��m����T��b�S�e�a�maximal�ideal.�Then�dene��J�r�[�m�]�=��f�t��2����J�r�(����!�����Q���
#��)�j�xt�UR�=�0�8�x��2��m�g�.��Then��w�J��[�m�]����J��[��],���and��w�J��[��]�is�isomorphic�as�a�group�to�(�Z�=�Z�)����2�2�g��	�.�мJ��[��]���is��a��T�=�T�-mo�S�dule.������But��J�r�[�m�]�is�\b�S�etter"�b�ecause�it�is�a��T�=�m�-mo�dule�and��T�=�m��is�a�eld,��~so��J�r�[�m�]�is�a�v��rector���space.�8�J�r�[�m�]��is�a�represen��rtation�of��k�g�[�G�]�UR=�(�T�=�m�)[�G��.�al�C��(����!�����Q���
#��=�Q�)].������Naiv��re��Idea:�8�J�r�[�m�]�is�a�mo�S�del�for�������m���5�,�at�least�when�������m��
�ݹis�irreducible.�������Theorem��21���L���If�35��UR�6�=�2��and����-��N�@��,�then��J�r�[�m�]��is�a�mo��ffdel�for�������m���5�.������Note:�8�J�r�[�m�]��ma��ry�indeed�b�S�e�a�mo�del�for�������m��
�ݹin�other�cases.�������Let�꨼V��¹=�UR�k�ſ����k�QŹwith�the�������m���5�-action�of��G��.�al�C��(����!�����Q���
#��=�Q�).������F��Vact:��8�dim����(����T�=�m��,�t�H���V���2�0���Z�(�X���̺0����(�N�@�)���߽=�F��i?������;����
����2�1���)[�m�]�UR���1.�������52����5����K����b�{�K������{��8�z�Marc��u�h,�1996��b#��Scrib�S�e:�8�Da��rvid��M�Jones,��<[email protected]>��������T�UR�=��Z�[�:::T���̽n���P�:::�]�����End��b�J���̺0����(�N�@�).�WLet���m����T��b�S�e�a�maximal�ideal�with��m�j��.�WAs�w��re�sho�w�ed�last���time,��there�exists�a�represen��rtation���d���������m��
3��:�UR�G��.�al�C��(����!�����Q���
#��=�Q�)��!���GL������̺2����(�T�=�m�)���whic��rh��is�semisimple.���N����Denition��12���Ra��L��ffet���R��^�b�e�a��Z���̽�����-algebr�a.�G^Then�we�say�that��R��^�is�Gor�enstein�if���Hom���O�����Z��i?����&i:�(�R�J;����Z���̽����)�����P���UR����԰���n:�=���������R�J�.������Goal:�8�T��Vo��pro��rv�e�that�if��m�UR�-��2�N�+��and�������m��
�ݹis�irreducible,�then��T�����m���is�Gorenstein.������Theorem��22���L���If���J�g��=�K�J���̺0����(�N�@�)�;���J�r�[�m�]����J��[��]�,��then����dim���M8����T�=�m��,<��J��[�m�]�=�2�.��That���is,��������m���%�arrises�fr��ffom����J�r�[�m�]�.������Remarks:��-Let��N�M��*�=��OFHom����(�T�;����Z�)�OF=�(�S���̺2����(���̺0���(�N�@�))�;��Z�),�8a��N�T�-mo�S�dule.�
w�Sa��rying�that��M�����m�����is���Gorenstein��amoun��rts�to��M�����m��
�ݹb�S�eing�free�of�rank�1�o�v�er��T�����m���5�.������Let�#>�W�&�=�i�J�r�[�m�].��W��Ve�will�compare��W���and��V��p�,�qcwhere��V����is�the�Galois�mo�S�dule�giv��ren�b�y���������m��
3��:�UR�G��.�al�C��(����!�����Q���
#��=�Q�)��!���Aut���=����T�=�m��),e�V��p�.������Let�꨼W���Ɵ��2�ss����b�S�e�the�direct�sum�of�the�Jordan-Holder�quotien��rts�of��W��ƹ.���z����Theorem��23���L���(Mazur)�>�W���Ɵ��2�ss����is�isomoprhic�as�a�gr��ffoup�to��V�ƿ�pV�:::����V��p�,���with�>�t��c�opies�of��V��p�.���(This�35is�true��8�N�t�as�long�as����is�irr��ffe�ducible.)������Remark:�8�A��v��X�arian��rt�of�this�is�the�follo�wing���N����Denition��13���Ra��A�9:r��ffepr�esentation�9~������m��:�:�;�G��.�al�C��(����!�����Q���
#��=�Q�)��!���GL���ʠ��̺2�����(�T�=�m�)��is�absolutely�irr��ffe�ducible�if���the�Klc��ffomp�osition��G��.�al�C��(����!�����Q���
#��=�Q�)�\8�!���GL����ӟ�̺2���׹(�T�=�m�)��,���!���GL����ӟ�̺2���(�����щfew�
3/��T�=�m����w�)�Kl�is�irr��ffe�ducible,��ywher�e���Kl���щfew�
3/��T�=�m����#
��is�Klthe���algebr��ffaic�35closur�e�of��T�=�m�.������Theorem��24���L���(Boston-L��ffenstr�a-R��2ib�et)��|If�������m��w��is�absolutely�irr��ffe�ducible,��then��|�W�;B�is�isomorphic���as�35a�gr��ffoup�to��V�G�����:::����V��p�.�������Pr���oof.����(Of��Mazur's�theorem)�Supp�S�ose��d�UR�=��dim��꘼W��ƹ.�8�Consider���d���tv�Hom�����u����F������=�(�W��r;������̽����)�UR=��Hom����۟���T�=�m��-�)�(�W�;����T�=�m�(1))�=��W���Ɵ�����aʼ:����W��Ve��will�sho��rw�that������?�W���Ɵ���ss��������(�W���Ɵ�����aʹ)�����ss��
���=�UR�V�G���:::����V������where��the�pro�S�duct�has�2�d��copies�of��V��p�.������W��Ve���will�sho��rw�that�the�t�w�o�represen�tations�ab�S�o�v�e�ha�v�e�the�same�c�haracteristic�p�S�olynomials.���Then��the�Brauer-Nesbitt�theorem�will�pro��rv�e��that�they�are�isomorphic.�������53����6�m��K����b�{�K������{����Lo�S�ok��Bat��frob���z4��̽p��A��.�7�It�has�the�follo��rwing�c�haracteristic�p�S�olynomial�on��V��p�:�8�X�����2�2��
6��X��T���̽p���]�X�J2�+��p�c�=����(�X�<����r�S��)(�X����pr��S����2��1���
�)�w|�2��(�T�=�m�)[�X��].�8JSo��!the�c��rharacteristic�p�S�olynomial�of��frob���%��̽p��"���on��V���p���2�d��9�is�(�X�<�����r�S��)����2�d��ߨ�(�X��+�����pr�����2��1���
�)����2�d���.������And��Tthe�c��rharacteristic�p�S�olynomial�of��frob����F��̽p��!Q��on��W���is�(�X�����^����̺1����)����:::����(�X�������̽d��ߨ�),��where��T����̽i��b.�is����r����or�E'�pr��S����2��1���
�.�H\The�����̽i��dڹ's�are�of�this�form�b�S�ecause��frob�����W����2��"����p��� �����E�T���̽p����]�frob���WO��̽p��!�+��p��[�=�0.�So�E'the�c��rharacteristic�����p�S�olynomial�Geof��frob����W��̽p��!��on��W������̼W���Ɵ��2���
�/�is�(�X��O������̺1����)(�X����p���������1�����1���p�)����:::����(�X�������̽d��ߨ�)(�X����p���������1���%��d���p�)��,=�(�X��O�����r�S��)����2�d��ߨ�(�X��+�����pr�����2��1���
�)����2�d���.�{�����������Theorem��25���L�޼J�r�[�m�]�UR�6�=�0�.�fiie.��t�UR���1�.�������Pr���oof.��Supp�S�ose�]S�J�r�[�m�]��>=�0.�	��Then��J�r�[�m����2�i��dڹ]�=�0�8�i����1.�	��This�follo��rws�from�Nak��X�a�y�ama's����lemma.�n�So�Q�no��rw�lo�S�ok�at�the���-divisible�group��J�����m��
�\�=�'�[���̽i��dڼJ�r�[�m����2�i���].�So�Q�in�order�to�sho��rw��J�r�[�m�]�'�6�=�0,���equiv��X�alen��rtly��V,�́w�e���will�sho��rw�that��J�����m��
3��6�=�UR0.�,PF�rom�the�discussion�to�follo��rw,�́w�e���will�see�that��J�����m��
�-�is���dual�P
to��T���a����2����RA��m����5�J�r�.�iBut��T�a����2����RA��m����5�J�l|�is�non-zero�b��ry�the�theorem�of�W��Veil�stated�b�S�elo�w,�iband�th�us��J�����m��.?�is���non-zero;��hence,��J�r�[�m�]�is�non-zero.������������Theorem��26���L���(Weil)�35�T����
��Z���̽��N8���UR�End��b�J���
��Z���̽��,�inje��ffcts�into���End��?�(�T���a���̽����J�r�)�.������Corollary��2���H��T���a���̽����J�qĿ6�=�UR0�.����Pr���oof.��"�Since��Z�T��L�
��Z���̽��cӿ6�=�j�0�and��T��
��Z���̽���@�injects�in��rto��End��(�T���a����2����RA�������J�r�),���th�us��End��(�T���a����2����RA�������J�r�)�j�6�=�0�and�so����w��re��m�ust�ha�v�e��T���a����2����RA�������J�qĿ6�=�UR0.�N���������No��rw��w�e�will�discuss�the�ab�S�o�v�e�dualit�y�b�S�et�w�een��J�����m��
�ݹand��T���a����2����RA��m����5�J�r�.������T���a���̽����J���=�����rlim�����f� ������T��������,�J�r�[�����2�i��dڹ],��kwhic��rh���is�isomorphic�as�a�group�to��Z����2��2��j�dim���J��RA����%�.�u�On�the�other�hand,��J���̽���X�=�����[����2��1��RA��i�=1���AV�J�r�[�����2�i��dڹ]��whic��rh�is�isomorphic�as�a�group�to�(�Q���̽����=�Z���̽���)����2�2��j�dim���J��%�.��������Denition��14���Ra��The�35c��ffovariant�T���ate�mo�dule��T���a���̽����J�O��is���Hom����(�Q���̽���=�Z���̽���;���J���̽���)�.������Denition��15���Ra��The�35c��ffontr�avariant�T���ate�mo�dule��T���a����2����RA�������J�O��is���Hom����(�J���̽����;����Q���̽���=�Z���̽���)�.������The��con��rtra�v��X�arian�t�T��Vate�mo�S�dule�is�the�Pn�try�a��jin�dual�of��J���̽����.�������Note��that��Hom��d1(�J���̽����;����Q���̽���=�Z���̽���)�UR=��Hom����۟���Z��i?����%�x�(�T���a���̽����J��:;��Z���̽���)��so��T���a����2����RA�������J�qĹ=��URHom����(�T�a���̽����J��:;��Z���̽���).������No��rw,��w�e��Bha�v�e�the�W��Veil�pairing,��J�r�[�����2�i��dڹ]��s���J��[�����2�i��dڹ]�TM�!�����������i���%X�.�
��P��rassing��Bto�the�limit�induces�a���pairing��<;���>�:�J$�T���a���̽����J��n��r��T�a���̽���J�f��!�J$�Z���̽���(1)�=����lim������ ������T��������޼���������i���%X�.���This��pairing�is�p�S�erfect�and�giv��res�us�that��m'��T���a���̽����J����������7���g	��r�!������
t�Hom��%��(�T�a���̽���J��:;����Z���̽���(1))�UR=�(�T���a����2����RA�������J�r�)(1).������No��rw�tUif��t�?��2��T�,���then��tU�h��	�tx;���y��n9�i��	X��=���h���T�x;�t����2�_��*��y��n9�i��	�9�where�tU�t����2�_��
jY�=��w�R�tw����and��w��S�=��w���̽N��X�2���End��Lh�J���̺0����(�N�@�)�is�the���A��rtkin-Lehner��in�v�olution.�8�So�this�pairing�is�not��T�-compatible.������So��w��re�dene�a�new�pairing,�whic�h�is��T�-compatible,�b�y��f���s�[�꨼;�ꦹ]�UR=���h�����x;���w�R�y��n9�i���:����Chec��rk��for��T�-compatibilit�y:�8�[�tx;���y�n9�]�UR=���h�����tx;�w�R�y��n9�i��n6�=���h�����x;�t����2�_��*��w�R�y��n9�i��n6�=���h�����x;�w�R�ty��n9�i��n6�=�[�x;�ty�n9�].�������As��a�reminder:�8�w�=R�is�an�in��rv�olution��on��X���̺0����(�N�@�)�giv��ren�b�y�(�E��;���C�ܞ�)�UR�7!��(�E�=C�5�;���E��[�N�@�]�=C�ܞ�).������[�꨼;�ꦹ]�denes�an�isomorphism�of��T����
��Z���̽����-mo�S�dules:��f���:|�T���a���̽����J����������7���g	��r�!�����
t�T�a���������ڍ������J�r�(1)�UR=��Hom����۟���Z��i?����%�x�(�T���a���̽���J��:;����Z���̽���(1))���whic��rh��is�isomorphic�as�a�group�to��Hom��d1(�T���a���̽����J��:;����Z���̽���)�UR=��T�a����2����RA�������J�r�.�������1�������54����7	A��K����b�{�K������{��Marc��u�h�z�11,�1996��b#��Scrib�S�e:�8�J�� anos��Csirik,��<[email protected]>�������Our��Wbasic�situation�is�that�w��re�ha�v�e��J���̺0����(�N�@�)�(or��J���̺1���(�N�@�))�and�a�maximal�ideal��m�����T��������End���(�J���̺0����(�N�@�))�lying�o��rv�er���.��-W��Ve�ha��rv�e��T���
�����Z���	F�Z���̽����=��������Q���
����m�j�����J�T�����m���,�,�"3where���T�����m���o�is�the��m�-adic�completion���of�jx�T�.��OThis�ring�acts�on���T��Va����6��̽�����,��lbreaking�it�up�as�������Q���
�A���m�j������ιT��Va���)����m���0���.�The�last�time�w��re�ha�v�e�conrmed���that����T��Va���Lf����m����6�=�UR0,��and�that���T��Va���Lf��̽���/�is��T����
���Z���̽���㎹-auto�S�dual,��hence�eac��rh���T�����m����is���Z���̽���#��-auto�dual.������On���a�nite�lev��rel,��tak�e���J�r�[�m�]�UR=��W��ƹ,�a���represen��rtation�of��G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)�o�v�er�the�eld��T�=�m�.�JAlso����Z���V��¹=��UR2-dimensional��represen��rtation�of����N�G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)���o��rv�er���^�T�����m������whic��rh�x�giv�es�the�represen�tation�������m��
W �(whic�h�is�giv�en�b�y�its�trace,���determinan�t�and�the�fact�that���it��is�semisimple.)������No��rw��assume�that��V���is�irreducible.�8�Then�����4��W���Ɵ���ss������P���
S�����԰���
lx�=������9�V�G�����������UN����V����(with��ݼt��copies�of��V�XM�on�the�righ��rt�hand�side).���This�w�as�pro�v�ed�in�the�last�lecture�using�an���argumen��rt��of�Mazur.�8�W��Ve�w�an�t�to�sho�w�that��t�UR�=�1.������W��Ve�<�already�kno��rw�that��t�UR>��0,�_�since�<�b�y�Nak��X�a�y�ama's�Lemma,�_��W���=�UR0�implies���T��Va���������m����C�=�0�whic��rh���w��re��kno�w�is�false.������Ho��rw�K�is�one�supp�S�osed�to�think�ab�out��J�r�[��]?�\$Think�of�it�as��J��[��]���=���T��Va���\R��̽���U8�=���T��Va��� a���̽���Z��.�\$Since���K�T��Va�����x���}��������}������@�is���the��isomorphic�linear�dual�of���T��Va���Lf��̽���EL�,�y��rou�can�think�of��J�r�[��]�as���T��Va�����x��Lf�������Lf�����j�=���T��Va�����x��
a��������
a������!¹.�8�Then�������J�r�[�m�]�UR=���T��Va�����x������������m�����E�=�m���T��Va�����x��
a��������
a��m�������b��ry��P�on�trjagin�dualit�y��V.������More�p�on�P��ron�trjagin�p�dualit�y:�Eb�J���̽��2��=�9ּJ�r�[�����2�1��	�]�is�P�on�trjagin�dual�to���T��Va�����x��ҧ�������ҧ�����́�=��9�Hom����_����Z��i?����&���(��T��Va���
a���̽���Z��;�����Z���̽���8�)�9�=����Hom���y�����Z��i?����"�&�(�J�r�[�����2�1��	�]�;�����Q���̽���q�=��Z���̽���8�).������In��general,�0NP��ron�trjagin�dualit�y�assigns�to�eac�h�discrete�ab�S�elian�group��M�B��a�compact�ab�elian���group�2��M��@���2���	V:�=��URHom����(�M���;����Q�=�Z�);�o�and�to�eac��rh�compact�ab�S�elian�group��N�sx�a�discrete��Hom���(�N�;����Q�=�Z�).���Therefore��the�groups��Z��and��Q�=�Z��corresp�S�ond�to�one�another.������W��Ve�{are�using�a�kind�of�lo�S�cal�v��rersion�under�whic�h�the�group���Q���̽����=��Z���̽���<a�is�sen�t�to���Z���̽���?ܹand�the���group����Z���̽���6�is��sen��rt�to���Q���̽����=��Z���̽���8�.������Giv��ren��the�group��������J�r�[������1��	�]�UR=��Hom����(��T��Va�����x��
a��������
a������!¼;�����Q���̽���q�=��Z���̽���8�)�;����it��has�the�subgroup�������J�r�[�m�����1��	�]�UR=��Hom����(��T��Va�����x��
a��������
a��m����?�;�����Q���̽���q�=��Z���̽���8�)�;����whic��rh��in�turn�con�tains�������J�r�[�m�]�UR=��Hom����۟���Z�=�Z��0�(��T��Va�����x��
a��������
a��m����?�=�m���T��Va�����x��
a��������
a��m�����;�����Q���̽���q�=��Z���̽���8�)�:��������55����8���K����b�{�K������{��Since��the�part�of���Q���̽����=��Z���̽���#��killed�b��ry����is�just��Z�=�Z�,�this�can�b�S�e�written�as���A���(z�J�r�[�m�]�UR=��Hom����۟���Z�=�Z��0�(��T��Va�����x��
a��������
a��m����?�=�m���T��Va�����x��
a��������
a��m�����;����Z�=�Z�)�:����No��rw��if��J�r�[�m�]�is�zero,�1then�so�m�ust�b�S�e���T��Va�����x��dh�������dh�m����B��=�m���T��Va�����x��
a��������
a��m����?�,�1but�then�b�y�Nak��X�a�y�ama's�Lemma���T��Va�����x��dh�������dh�m����EG�should����b�S�e��zero,�whic��rh�it�isn't.���%����W��Ve��ha��rv�e�the�follo�wing�to�goals:�������First�go���al.�N�T��Vo�p�sho��rw�that��t�UR�=�1,��Ii.e.,�that�p�J�r�[�m�]�is�generated�b�y�t�w�o�elemen�ts�o�v�er��T�=�m�.���(Since��F�J�r�[�m�]�is�killed�b��ry��m�,�ҭthis�is�equiv��X�alen�t�to�sho�wing�that��J�r�[�m�]�is�generated�o�v�er��T��b�y���t��rw�o��elemen�ts.)�������Second���go���al.�8�T��Vo��sho��rw�that���T�����m����is�Gorenstein,�i.e.,�that���A����c��T�����m�������P����������԰������=������ҥLHom����՟���Z��i?�����8r�(��T�����m���A¼;�����Z���̽���8�)�:����(Wh��ry�V<w�ould�w�e�w�an�t�this?�{�W��Vell,�q!recall�that��S��׹(�Z�)�����P���n����԰���%V�=������n3Hom���)缟���Z��0d�(�T�;����Z�)�as��T�-mo�S�dules.�Our�goal����just��sa��rys�that��S��׹(�Z�)�is�lo�S�cally�free�at��m�.)������W��Ve�M�are�assuming�that�������m��+��is�irreducible,�f�and�that�����6�j�2�N�@�.�b6���6�j�N����means�M�that��J�j7�has�go�S�o�d���reduction��at���,�and�2�UR�6�j�N�+��means��that�w��re�can�apply�Ra�ynaud's�results�on�group�sc�hemes.������P��rostp�S�one�<9pro�ving�the�rst�goal�no�w,���and�sho�w�instead�wh�y�the�rst�goal�implies�the���second��one.������By��Nak��X�a��ry�ama's�Lemma,����T��Va���Ve����m���'A�is�generated�b�y�2�elemen�ts�as�a�mo�S�dule�o�v�er���T�����m���4i�.�P�W��Ve�need���that����T��Va�����x��Lf�������Lf�m����*��=�m���T��Va�����x��
a��������
a��m�����is��also�generated�b��ry�2�elemen�ts:�8�but�for��T�=�m��v�ector�spaces��R�J�,�w�e�ha�v�e���A������Hom����F����Z�=�Z���B��(�R�J;����Z�=�Z�)�����P���UR����԰���n:�=��������Hom���(y�����T�=�m��:hҹ(�R�;��Z�=�Z�)���b��ry��considering�the�trace.�8�So�w�e�can�nd�a�map������2|�T�����m����������T�����m���A�����UR�T��Va��������m����E�:��
j����Claim��5���������2�rank���,b����Z��i?�����!E��T�����m���4��=��URrank���������Z��i?������$�Q�T��Va���1�����m���8�D�:��hJ����Recall���that��rank��� ]����Z��i?������&0��T��Va���3����̽���<�<�=� �2����dim���B�J�&�=�2����dim���T��=�2�rank���,b����Z��i?����!E��T�Y-� ���Z���̽�����.�9�This���mak��res�the�claim����pausible,��but�isn't�a�pro�S�of.�8�The�pro�of�is�p�ostp�oned�for�no��rw.������Accepting�?the�Claim,�T�it�follo��rws�b�y�basic�graduate�algebra�(Math�250)�that���T�����m���ex����V�T�����m�������P��� �����԰���$��=���������T��Va���
a�����m���?�.�8�Recall��that���T��Va���Lf����m���C�is�auto-dual�in�the�follo��rwing�sense:���A�����e^T��Va���������m�������P����������԰������=������ҥLHom����՟���Z��i?�����8r�(��T��Va���
a�����m���?�;�����Z���̽���8�)�:����Hence����T�����m����������T�����m����is��self-dual�in�the�same�sense.�8�Do�S�es�it�follo��rw�that���T�����m����is�self-dual?�������Y��Ves.�qT�o��_do�this,�use�a�stupid�thing:�^Mcall�t��rw�o��_copies�of���T�����m���<��F���̺1��50�=��u,�T�����m����M�and��F���̺2���=��u,�T�����m�����.�qThen���w��re��ha�v�e�����<�F������������ڍ�1���	r�����F������������ڍ�2�����������	����g	��aʿ!�����a̼F���̺1��j����F���̺2����;��������56����9-ڠ�K����b�{�K������{��where��'S���2���
��denotes�'Staking�linear�duals�(�Hom����of�the�thing�to���Z���̽���9�).���Then��F����2�������RA��1���	5������F����2�������RA��2���
��is�free,�6~so��F����2�������RA��1�������is��pro��jectiv��re,�so�it's�free�(o�v�er�a�lo�S�cal�ring,�all�pro��jectiv�e�mo�S�dules�are).�������No��rw��Lgoing�bac�k�to�our�claim,���note�that��rank���خ����Z��i?�����#�K�T�����m���7�_�=��URdim���ꚟ���Q��i?����"c��(��T�����m���b'�
�����Z��i?�����
:�Q���̽���Vu�).���Also��rank���خ����Z��i?������#�K�T��Va���1T	����m���;���=����dim����H����Q��i?����0�(��T��Va���
a�����m���꛿
�����Z��i?������E�Q���̽���ู).��������T��Va������̽���(|N�is���gotten�b��ry�considering�the���-p�S�o�w�er�torsion�p�S�oin�ts�in�the�Jacobian,��zor�else�it�is���just��gthose�complex�p�S�oin��rts�in��J�r�(�C�).�0But�using�the�Albanese�co�v��X�arian�t�in�terpretation,��W(and���recalling��that��H���V���2�1���Z�(�X���̺0����(�N�@�)�;����C�)�UR=��S���̺2���(���̺0���(�N�@�)�;����C�)),��!;����=�J�r�(�C�)�UR=������ō����Hom�������C��$���(�S���̺2����(���̺0���(�N�@�)�;����C�)�;��C�)�����Qm�fe|:$���d��H���̺1����(�X���̺0���(�N�@�)�;����Z�)�������ܼ:��!;>����The��lattice��L�UR�=��H���̺1����(�X���̺0���(�N�@�)�;����Z�)��is�a��T�-mo�S�dule,�and���T��Va���Lf��̽������=�UR�L����
�����Z���	'��Z���̽�����.������Instead,��w��re'll�lo�S�ok�at������DId�L����
�����Z��	'��R�����u���=��������Hom���������C������(�S���̺2����(���̺0���(�N�@�)�;����C�)�;��C�)�UR=��Hom����۟���R��#�[�(�S���̺2����(���̺0���(�N��)�;����R�)�;��C�)���������u��=��������Hom���������R�����(�S���̺2����(���̺0���(�N�@�)�;����R�)�;��R�)����
�����R��
}(�C�UR�=�(�T��
�����Z��	'��R�)��
�����R��
}(�C�;������hence��E�L�¿
�����Z�����R�UR�=�(�T��
�����Z�����R�)��
�����R��	�B�C�,���so��E�L��
�����Z���R��E�is�free�of�rank�2�o��rv�er��E�T��
�����Z�����R�,���therefore��L��
�����Z�����C����is��free�of�rank�2�o��rv�er���T����
�����Z��	'��C�.������No��rw�3��cho��ffose��an�em�b�S�edding���Q���̽���!W�,���!��\�C�.���T��Va���u=��̽���?�=�������Q���
C%���m����!X�T��Va���#�����m���.�ӹis�3�a�mo�dule�o��rv�er�3��T��F�
���Z���̽���戹=���\�����Q���
C%���m��!X�T�����m���5�.���Consider����T��Va���Lf��̽�����
����Z�Q���̽���qŹ=��UR�����Q�������m���P�(��T��Va���
a�����m���꛿
�����Z��i?������E�Q���̽���ู.�����������Q�������m��!��(��T��Va���
a�����m���+��
�����Z��i?�����2�Q���̽���!��)�w�=���T��Va������̽����ۿ
�����Z��i?����2�C��=��L�땿
�����Z��
h��C���is�free�of�rank�2�o��rv�er���T��
�����Z��
h��C�w��=�(�T��
�����Z������Z���̽���8�)��ѿ
�����Z��i?����
n�C�뷹=�������Q���
]����m��;��(��T��Va���
a�����m���&Ŀ
�����Z��i?�����C�).�A�Hence���CT��Va�������m���iȿ
�����Z��i?�����C�C�is�free�of�rank�2�o��rv�er�C�T�����m��	��
�����Z��i?�����C�,�Ytherefore����dim����H����C����)3�T��Va���*�����m���4Ւ�
�����Z��i?����
�	�C�9��=�2����dim����F����C��)1�T�����m��
J��
�����Z��i?�����C�,�Ngso��@dim���������Q��i?������&n�T��Va���3w,����m���=�Ϳ
�����Z��i?������Q���̽��� �8�=�2����dim����F����Q��i?����#,�T�����m��
J��
�����Z��i?������Q���̽����|�,�Ngas�@the���dimension��of�a�v��rector�space�do�S�es�not�c�hange�if�w�e�tensor�up�to�a�bigger�eld.������This�8settles�our�claim�that�the�second�goal�implies�the�rst�one.� �The�rst�goal�shall�b�S�e���pro��rv�ed��in�the�next�lecture.��(V��Marc��u�h�z�13,�1996��b#��Scrib�S�e:�8�Amo�d��Agashe,��<[email protected]>������Let���S�p�b�S�e�a�sc��rheme.��+Then�a�group�sc�heme�o�v�er��S�p�is�a�group�ob��ject�in�the�category�of����S��׹-sc��rhemes��i.e.��6it�is�an��S��-sc��rheme�G�pwith��S��-morhphisms��m�UR�:��G�����G�UR�!��G;���i��:��G��!��G;�e��:��S�)�!��G����satisfying��the�usual�group�axioms�(See�[1]�for�details).������W��Ve�^\will�b�S�e�in��rterested�in�nite�
at�group�sc�hemes�o�v�er��F���̽p��	%��where��p��is�a�prime�and��F���̽p���is���the���nite�eld�with��p��elemen��rts.�1�Let��A��b�S�e�a��F���̽p���]�-v�ector�space�and�let��G�UR�=��S���pec�(�A�).�1�Then���the���order�.of��G�,���#�G�UR�=��dim�����F���p���A�A�.�
If�.�R��x�is�an��F���̽p���]�-algebra,�then��G�(�R�J�)�UR=��M�@�or�S��(�S���pecR�;���G�)�.is�the�group���of��R-v��X�alued�p�S�oin��rts�of��G�.������Here��are�some�examples:���1)The��additiv��re�group�sc�heme����̽a��Y!�=�UR�S���pec�(�F���̽p���]�[�T��ƹ]).�������57����:<à�K����b�{�K������{��2)��The�m��rultiplicativ�e��group�sc��rheme����̽m��Z�=�UR�S���pec�(�F���̽p���]�[�T���;���T���Ɵ��2��1���B�]).����3)��The��p�th�ro�S�ots�of�unit��ry�����̽p����is�the�k�ernel�of�the��p�-p�S�o�w�er�map�on����̽m��Ĺ.���4)��The�group�����̽p����is�the�k��rernel�of�the�F��Vrob�S�enius�map�on����̽a��Ϲ.���5)�J Let��E��=�F���̽p��	}�b�S�e�an�elliptic�curv��re�though�t�of�as�a�group�sc�heme.�WIThen��E���[�p�]�is�the�k�ernel�of���the��m��rultiplication�b�y��p��map�on��E���.�8�It�has�order��p����2�2�����and��U]��E���[�p�](�����fe
Cϟ	n��F���̽p����
CϹ)�UR=�����f\�(�����d���c��1���� I:if��E�is�sup�S�ersingular��������c��p���� I:�if��E�is�ordinary�������
����The���primary�reference�for�what�follo��rws�is�[2].�6�Let��J�qĹ=�UR�J���̺0����(�N�@�)�b�S�e�the�Jacobian�of��X���̺0���(�N�@�).���Then�[��J�x�is�dened�o��rv�er�[��Q��and�has�go�S�o�d�[�reduction�at�primes�not�dividing��N�@�.�	.Let��l�� �b�S�e�a�prime���not���dividing��N�@�.��Then��J�r�[�l�C��]�extends�to�a�nite�
at�group�sc��rheme�o�v�er��Z�[�����Fu���Ӻ1��33�콉fe�D��'��N�����	���]�(Ref:�'Q[3,��exp.���IX]).��This�in�turn�giv��res�rise�to�a�group�sc�heme�o�v�er��F���̽l��p�whic�h�can�b�S�e�though�t�of�as��J���߽=�F��i?�l���
�R�[�l�C��].������Let���V�dx�b�S�e�the�t��rw�o��dimensional�v��rector�space�o�v�er��T�=�m��whic�h�giv�es�rise�to�the�represen�tation���������m��
�(�constructed���last�w��reek.�?�W��Ve�w�an�t�to�sho�w�that�this�is�isomorphic�to�the�naturally�dened���Galois�represen��rtation��W���=�@+�J�r�[�m�].���W��Ve�ha�v�e�0�@+���V�ܛ���W�����J�r�[�l�C��].���W��Ve�mak�e�the�follo�wing���assumptions:�&Z�l���6�URj�N���;���l��6�=�2�Ŝand��V�b�is�irreducible.�,�This�is�b�S�ecause�w��re�will�b�e�in��rv�oking�Ŝtheorems���of��Ra��rynaud.������Let�����J�������fe���1�denote���J��t�considered�as�a�group�sc��rheme.���Then����J������fe���f�[�l�C��]�is�a�nite�
at�group�sc�heme���o��rv�er����Z���̽l��!ȹ;�omin�fact,�.�it�is�the�Sp�S�ec�of�a�free�nite�rank�mo�dule�o��rv�er����Z���̽l����of�rank��l��C����2�2�g��	X��.�BkRa��rynaud���sho��rw�ed�x(See�[4]�or�[1])�that�if��l���6�=��^2�(so�that�the�absolute�ramication�index����=�1��<�l�տ��F�1),���almost���ev��rerything�ab�S�out�the�group�sc�heme����J������fe���S��[�l�C��]�can�b�S�e�seen�in�terms�of�its�Galois�mo�dule����J�r�[�l�C��](�����fe
EU�	n��Q���̽l����
EU�).��Let���z�V��z����fe	v��kӹand���z�W��z����fe�����_�b�S�e�z�the�group�sc��rhemes�giv�en�b�y�the�Zariski�closure�of��V�W�and��W���in������J�����fe�����[�l�C��]���߽=�Z��i?�l������resp�S�ectiv��rely��V.�8�Then��w�e�ha�v�e:���������4C�V���4C����fe	v���������UR�W��UR����fe����d5����UR�J��UR����fe���
�V�[�l�C��].������Our��Bgoal�is�to�sho��rw�that��V���,���!�UR�W�8�is�an�isomorphism.��Ra�ynaud�sho�w�ed�that�the�category����of�mfnite�
at�group�sc��rhemes�o�v�er��Z���̽l���.�is�an�ab�S�elian�category��V,��so�it�mak�es�sense�to�talk�ab�S�out���the�Vquotien��rt����Q�����fe	NW�����=���0�W��0����fe������=��V�����fe	v��	v�,�p�whic�h�is�again�a�nite�
at�group�sc�heme�o�v�er��Z���̽l��!ȹ.�{/Note�that����Q�����fe	NW���
�o��M�=�Q��i?�l�����M��corresp�S�onds��vto��W�=V��p�.�SJTh��rus,����V���=�dO�W��,����Q�����fe	NW�������M�=�Q��i?�l���S�=�0.�Since����v�Q���v���fe	NW��5C�is��v
at,���it�has�the�same�rank�o��rv�er��*���Q���̽l��p�as��o��rv�er��F���̽l��!ȹ.�8�P�assing�to�c�haracteristic��l�.7�yields�the�follo�wing�exact�sequence:���������0�UR�!����V������fe	v����W���=�F��i?�l������!����W������fe��������=�F��i?�l���!3��!����Q�����fe	NW�������M�=�F��i?�l����M�!��0.���_�Again,��since�the�ranks�remain�constan��rt,����(���V���,���!�UR�W��n�is��an�isomorphism��,����V������fe	v����W���=�F��i?�l������,���!����W������fe��������=�F��i?�l���!�ݹis�an�isomorphism��,����Q�����fe	NW�������M�F��i?�l����I�=�0.���J�F��Vollo��rwing�n{Ra�ynaud,����Q�V���Q����fe	v���V�,����Q�W���Q����fe�����]�and����Q�����fe	NW��+M�inherit�the�action�of��k��o�=�UR�T�=�m��(i.e.�|they�are��k�g�-v�ector�space����sc��rhemes).�8�W��Ve��will�sho�w�that����Q�����fe	NW���
8���M�F��i?�l�����=�UR0.�8�F��Vor�that,�w�e�need�Dieudonn���s�e�mo�S�dule�theory��V.��>�����Let�f��G=k��ʹb�S�e�a�nite��k�g�-v��rector�sc�heme�where��k��ʹis�a�nite�eld�of�order��q�n9�.��Supp�S�ose�the�order���of���G��is��q��n9���2�n����.� LThen�the�Dieudonn��r��s�e�mo�S�dule�functor��D��z�(See�[5])�is�a�con�tra�v��X�arian�t�functor�whic�h���maps�P�G��to�a��k�g�-v��rector�space��D�S��(�G�)�of�dimension��n�,�ikcalled�the�Dieudonn���s�e�mo�S�dule�of��G�.�iLet����F���r�S�ob�UR�:��G��!��G����b�e�the�morphism�on��G��induced�b��ry�the��p�th�p�o��rw�er���map�on�the�underlying�sets���and�PVlet��V��per���b�S�e�the�dual�of��F���r�ob�.�oThen�the�maps����and����induced�on��D��(�G�)�b��ry��F���r�ob��and��V��per��������58����;Oh��K����b�{�K������{��resp�S�ectiv��rely���satisfy���>������=�UR��w�����=�0.�'A�D�	Z�is���a�fully�faithful�functor�i.e.�w��re�can�tell�ev�erything����ab�S�out��the�group�sc��rheme�from�its�Dieudonn���s�e�mo�S�dule.������F��Vor�ȳexample,��}let��k��o�=�UR�F���̽l��w�=��F���̽p���]�.�-�Let�ȳ�G��=�����̽p���,��}����̽p����or�ȳ�Z�=p�Z�.�-�Then�in�all�cases,��dim���̽k��#��(�G�)�UR=�1.���F��Vor�꨼���̽p���]�,���UR�=����=�0;��for�����̽p���,���UR�=�0�;�����=��id�꨹and�for��Z�=p�Z�,���UR�=��id;���=�0.������Let��@�G����2�_��V͹=�,�H��Vom�(�G;������̽p���]�)�denote�the�Cartier�dual�of�G.�Then��D�S��(�G����2�_��*��)�=��H��Vom���̽k��#��(�D��(�G�)�;���k�g�)���with�꨼��and����o�in��rterc�hanged.������Here��is�a�sophisticated�example�of�the�ab�S�o��rv�e��theory:���Let��A��b�e�an�ab�elian�v��X�ariet��ry�o�v�er����F���̽l����and��׼G��=��A�[�l�C��].�HmThen,�1#�G��is�an��F���̽l��!ȹ-v��rector�space�sc�heme�of�order��l��C����2�2�g��Hh�and�th�us��D�S��(�G�)�has���dimension��2�g�n9�.�8�F��Vurthermore,��D�S��(�G�)�UR=��H����2���V�1��RA��D�<rR���)7�(�A���߽=�F��i?�l���
�R�).�Consider��the�Ho�S�dge�ltration:��f��l�e0�UR�!��H���V���2�0���Z�(�A;����
����2�1����)��!��H����2���V�1��RA��D�<rR���)7�(�A���߽=�F��i?�l���
�R�)��!��H���V���2�1���(�A;����O���̽A����)��!��0.������No��rw��H���V���2�1���Z�(�A;����O���̽A����)�UR=��T���an�(�A����2��_��RA��F��i?�l����	�N�)�=��H��Vom�(�H�����2�0���(�A����2�_��*��;����
����2�1����)�;��F���̽l��!ȹ)�and�it�corresp�S�onds�to�the�subgroup�����D�S��(�A�����F��i?�l���	�N�[�V��per��])��of��D�S��(�A���߽=�F��i?�l���
�R�)�where��V�er�>6�is�the�dual�to��F���r�S�ob�.������Consider��the�exact�sequence��f���B�0�UR�!��W���߽=�F��i?�l���$��!��J���߽=�F��i?�l��� �R�[�l�C��]��������&9�m��g ���!�������J���߽=�F��i?�l����[�l�C��].���Since�꨼D�>6�is�an�exact�functor,�applying��D��to�the�ab�S�o��rv�e,��w�e�get�the�exact�sequence�����m��D�S��(�J���߽=�F��i?�l��� �R�[�l�C��])��������&9�m��g ��UR�!�������D��(�J���߽=�F��i?�l����[�l�C��])�UR�!��D�S��(�W���߽=�F��i?�l����)��!��0.���In��other�w��rords,��D�S��(�W���߽=�F��i?�l��� �R�)�UR=��D��(�J���߽=�F��i?�l��� �R�[�l�C��])�=�m�D��(�J���߽=�F��i?�l����[�l�C��]).�������Next�T^time,�n�follo��rwing�F��Von�taine,�n�w�e�shall�consider��D�S��(�W���߽=�F��i?�l��� �R�[�V��per��])� A=��H���V���2�1���Z�(�J��:;����O�UV�)�=�m�H���V���2�1���(�J�;����O�UV�).���The��olatter�is�dual�(in�the�sense�of��H��Vom��of�a�v��rector�space�in�to��F���̽l��!ȹ)�to��H���V���2�0���Z�(�J�� r���2�_��G"�;���� ����2�1����)[�m�]�7�=����H���V���2�0���Z�(�X���̺0����(�N�@�)���߽=�F��i?�l��� �R�;���� ����2�1���)[�m�].���F��Vor��the�dualit��ry�ab�S�o�v�e,�L�w�e�need�to�use��J�� r���2�_�� ~��=�7k�Al�C�b�(�X���̺0����(�N�@�))�i.e.����J�Sݹ=����P���ic����2�0����(�X���̺0���(�N�@�)).� ��W��Ve�a9will�b�S�e�using�the�q-expansion�principle�to�sho��rw�that��D��(�W���߽=�F��i?�l��� �R�[�V��per��])�is���small.��V� �����References:������[1]�8xS.�Shatz,���Group�Sc��rhemes,�F��Vormal�Groups,�and�p-Divisible�Groups.� "PIn��A��2rithmetic���Ge��ffometry�,��G.Cornell�and�J.Silv��rerman,�eds.,�Springer-V��Verlag,�New�Y�ork,�1986,�pg.�8�29-78.������[2]� �B.Mazur,��Mo�S�dular�Curv��res�and�the�Eisenstein�Ideal,��Publ.Math.���I.H.E.S.�,��47��(1977),���33-186.������[3]���S��r��s�eminaire�de�G���s�eom��etrie���alg��ebrique���du�Bois-Marie,� 67-69.� r�P��V.Deligne�and�N.Katz,����L��ffe�ctur�e�o�Notes�in�Mathematics�,�<�nos.���288,340�,�Berlin-Heidelb�S�erg-New�,cY��Vork,�Springer,�1972,���1973.������[4]�HMM.�Ra��rynaud,���Sc�h���s�emas�HMen�group�S�es�de�t��ryp�e�(p,p,...,p),����Bul���l.� *�So��ffc.�Math.�F���r.�,��102����(1974),��241-280.������[5]� T.Oda,�/The� rst�De�Rham�cohomology�group�and�Dieudonn��r��s�e�mo�S�dules,��A��2nn.���scient.������)���x�E.Norm.Sup.�,��4����2�e��\�s��r��s�eie,�t.�8��2��(1969),�63-135.��� ����59����<d��K����b�{�K������{��April�z�1,�1996��b#��Scrib�S�e:�8�J�� anos��Csirik,��<[email protected]>��+J���Lo�cal��prop�erties�of������̽����@��Summary���of�all�w��re�kno�w,�� without�pro�S�ofs:�� ����̽���Eҹis�asso�ciated�to��f�G��,�� a�newform�of�w��reigh�t���2,�lev�el�����N�@�,���c��rharacter�d�"�%t�:�(�Z�=n�Z�)����2����x�!��C����2���$�(i.e.,�w��re�d�don't�necessarily�only�consider����̺0����(�N�@�)�an�y�more),���conductor�;[of��"��dividing��N�|?�(not�necessarily�equal�to��N�@�,�O�could�b�S�e�trivial).�*�Let���j��,�and�lo�S�ok���lo�S�cally��at��p��(with��p�UR�6�=���).������The��real�story�is�that��f�@�giv��res�rise�to�some�irreducible�represen�tation�of��GL����(2�;����A�),��twhic�h���corresp�S�onds�i|to�a�family�of�represen��rtations�(����̽v���
�),��Rwith�����̽v��V��b�eing�a�represen��rtation�of��GL���(2�;����Q���̽v���
�).���Here�ņ�v�3��sto�S�o�d�for�an��ry�v��X�aluation�of��Q�,���but�for�the�arc�himedean�prime�in�the�w�eigh�t�2�case�w�e���alw��ra�ys��fget�a�w��rell-kno�wn��fdiscrete�series�represen��rtation�on��GL��-(2�;����R�),��so�w�e�really�only�care���ab�S�out��the�represen��rtations�����̽p����on��GL��zC(2�;����Q���̽p���]�).������It�is�a�theoretical�result�of�Cara��ry�ol�(building�on�the�w��rork�of�Deligne�and�Langlands)�that��������̽����οj���̽D���p������up��to�isomorphism�only�dep�S�ends�on�����̽p���]�.������In���our�case,���w��re�are�most�in�terested�in�the�case�where��p����2�2��Z�6�URj�N�@�.���A��zreference�is�R.�P��V.�Langlands:����Mo��ffdular�35F���orms�and���-adic�r�epr�esentations�,��pp.�8�361{500�in�An��rt�w�erp��I�S�I.������F��Vor��2a�w��rarm�up,�*consider��2the�case�of��p�K�6�Gj�N�@�.�7~Then�w��re�ha�v�e�the�follo�wing�\c�haracteristic���p�S�olynomial"��
oating�ab�out:��ڼX�����2�2�������a���̽p���]�X����+��p"�(�p�).���F��Vactor��it�as�(�X�����r�S��)(�X����s�)��with��r��r;���s��п2��C�.���F��Vrom�[;the�Riemann�Hyp�S�othesis�for�ab�elian�v��X�arieties�(pro��rv�ed�[;b�y�Andr���s�e�W��Veil)�it�follo�ws�that����j�r�S��j�UR�=��j�s�j��=������ƿp���
UT���Ɖz�孟m:��p����;�.����������̽���$¿j���̽D���p����e�is� \an�unrami ed�represen��rtation�so��I���̽p�� й�is�killed�b�y�it,�Qso��frob����N��̽p��#j�mak�es�sense�up�to���conjugacy��V.��tSo���c����̽���$1�(�frob������̽p��WO�)��chas�a�w��rell-dened�c�haracteristic�p�S�olynomial��X�����2�2�������a���̽p���]�X����+��p"�(�p�)�UR�2��E���[�X��],���where����E����is�the�eld�generated�o��rv�er����Q��b��ry�the�co�S�ecien�ts�of�our�newform��f�G��.�\�So�since��p��do�S�es��U]�not��divide��N�@�,������̽���ov�(�frob������̽p��WO�)�is�semisimple,�so�it�follo��rws�that������̽����(�frob������̽p��WO�)�is�conjugate�to�����f\� �����d����Ƽr��� �P�0��������T0��� �޼s�����+�L��f\�!���3��.��U^����What��is�����̽p����supp�S�osed�to�b�e?�8�Let�������b���;�������:�UR�Q���������ڍ�p�����!��C����������b�S�e�8Gr�� ossenc��rharacters�(i.e.,�K_c�haraters�with�v��X�alues�that�don't�necessarily�ha�v�e�absolute�v��X�alue����1),��suc��rh�that��������\h1.����_���;����64�are���unramied:�{Z���j�����Z��������G��p����?ƹ=�gļ��O�j�����Z��������G��p�����=�1���(here�w��re�used�1�to�denote�the�unit�c�haracter).����_�Note��Kthat�this�condition�is�called�b�S�eing�unramied�b�ecause�b��ry�lo�cal�class�eld�theory��V,����_��Z����2����RA��p�����corresp�S�onds��to��I���̽p���in��G��.�al�C��(�����fe
#��	n��Q����
#���T�p����=�Q���̽p���]�);�������\h2.����_����(�p����2��1��\|�)�UR=��r�>6�and�꨼��O�(�p����2��1���)�=��s�.������By���the�relation�b�S�et��rw�een���Gr�� ossenc�haracters�and�c�haracters�of��G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)�whic�h�is�ex-���plained���in�W��Veil�(1950),��xor�J.-P�.�Serre�and�J.�T�ate,��x�Go��ffo�d�5Fr�e�duction�of�ab�elian�varieties�,��xAnn.�������60����=t_��K����b�{�K������{��of��Math.,��2�,�1968,�pp.�8�492{517,�or�Serre�(1972).����7�and������corresp�S�ond�to�c��rharacters�������C����̽��uZ�;������̽��ʬ�:�UR�G��.�al�C��(�����fe
#��	n��Q���
#��=�Q�)��!�������fe'��	fb��E���̽�����}Q������=U�;����whic��rh��satisfy��������\h1.����_����̽��uZ�;������̽��	�are��unramied;�������\h2.����_����̽��uZ�(�frob������̽p��WO�)�UR=��r�S��,�꨼���̽���(�frob������̽p��WO�)�=��s�.������Then��꨼���̽����ȹ=�UR����̽�� ��������̽��uZ�.��U]����Dene��a�c��rharacter�of�the�Borel�subgroup��B��X�=���UR��f\� �����d���?����� ?��������P�0��� ?�������[email protected]��f\�!���6��2��UR�GL����(2�;����Q���̽p���]�),��+���� ����:���UR��f\� �����d���?��x��� �Py����������0���!��z�����,/��f\�!���7V-�7!�UR����(�x�)���O�(�z���)�:��%L֍�This�uis�a�c��rharacter���烹:����B�.��!��C����2�����.��GThen�u����̽p��[R�=��Ind�����4����J�GL�� �J(2�;�Q���p��Z׺)��	;����J�B���:I���c�is�irreducible�since��r�S�=s��is�neither��p�����nor���p����2��1��\|�.�0LThis�is�called�the�(unramied)�principal�series�represen��rtation�attac�hed�to����{�and����O�.���(Unramied��in�this�case�since����7�and������are�unramied.)�8�This�is�denoted�����Ż�����̽p����=��URPS����(���;�����O�)�:������Here��is�the�end�of�the�w��rarm�up.������No��rw�#�lo�S�ok�at�the�case�where��p��exactly�divides��N�@�.��There�are�t�w�o�v�ery�dieren�t�cases,����according��^as��"��is�ramied�or�unramied�at��p�.��(It�is�ramied�i��p��divides�the�conductor�of��"�.)�������The�!�case�of��"��ramified�a��32t��p�.���Here�y^����̽p��	��=��H;PS����(���;�����O�),��and������̽���Fg�=�H;����̽���,��Ҽ���̽��uZ�.������(sa��ry)�is���unramied��and������is�ramied�(with�conductor�=��[�p�),��i.e.������O�j�����Z��������G��p����粹is�non��rtrivial.�But�it�is�not�to�S�o���bad��since����O�j���̺1+�p�Z��������G��p����Q-�=�UR1�for��p��6�=�2.������T��Vo��fsee�what's�going�on�(what�is������and����O�?),���think�ab�S�out��L�-functions.�~Let��f���=��������P��-P�a���̽n���P�q��n9���2�n�����and��x�atten��rtion�to��a���̽p����(whic�h�is�non-zero�b�y�the�analytic�p�S�oin�t�of�view).�8�By�denition,��!�ύ�=xc�L�(�f���;���s�)�UR=���������1��
���ԟ����X���ҁ���n�=1���Z�a���̽n���P�=n�����s��Î�=����N�����Y��������p�6j�N����g�(1������a���̽p���]�p������s��
�\�+��p"�(�p�)�p������2�s��J��)������1�����v�����Y�������
\z�p�j�N������(1����a���̽p���p������s��
��)������1��\|�;��&-O��the�꨼L�-factor�corresp�S�onding�to��p��b�eing�(1������a���̽p���]�p����2��s��
��)����2��1��\|�.������No��rw�N>�V���̽��
Ø�has�a�one-dimensional�quotien�t�on�whic�h��frob����0��̽p��#�˹acts�b�y�m�ultiplication�b�y�the���n��rum�b�S�er�qb�a���̽p���]�.��So�the�unramied�c��rharacter�����can�only�b�e�the�one�with�����(�p����2��1��\|�)�:�=��a���̽p���]�.��But�qbw��re���kno��rw���that�����̽��uZ����̽��ʬ�=��URdet��������̽���"�.�=�UR�"���̽�����(with��"��corresp�S�onding�to�the�Gr�� ossenc�haracter�taking��p����2��1��U�to����p�).�8�Hence��w��re�m�ust�ha�v�e����O�(�p����2��1��\|�)�UR=��"�(�p�)�pa����2���1��RA��p����.��������No��rw�use�the�form�ula����w}�fe��	����f���^�=��@O�����P���Έ���&�fe�Q�]ڍ�a���̽n�����ټq��n9���2�n��Vع=�@O�f���
�o
�"����2��1��\|�.��>F��Vor�(�n;���N�@�)�=�1,�S:w��re�ha�v�e��a���̽n��

���"�,�U�hence�@if�w��re�let�����̽��uZ�;������̽��	�ٹto�corresp�S�ond�to��f��~�and�����̽���;���
���̽��	�ٹto�corresp�S�ond�to����w}�fe��	����f���S&�,�U�w��re����ha��rv�e��	�����d2��f\� �����d����N޼���̽�����Y/�0���������*0����=q����̽�������T��f\�!���ߔ�=���UR��f\� �����d���?��
���̽��uZ�"���0qֹ0�������֜0���+M6����̽��uZ�"�����@vq��f\�!���Ja�;��A���so�$�up�to�p�S�erm��rutation�w�e�m�ust�ha�v�e��a���̽�� -a�=��� ���̽��uZ�"��and��b���̽���=������̽��uZ�"��(*).���No��rw�exactly�one�of�� ���̽�� ��and�������̽�� ��is��rami ed�(exactly�as�in�the�case�of�� �$%�and����O�).���Hence�b��ry�the�(*)�ab�S�o�v�e,�����̽��	��is�unramied���while�꨼
���̽��	�is�ramied.�8�So�using��det����=�UR����̽����"�,�w��re�get����q;����̽��ʬ�=�UR����̽��uZ�"��=��"���̽�������������1���%�����p�:����Lo�S�oking��at�the�F��Vrob�enii,�����&�fe
�^�]ڍ�a���̽p����2X�=�UR�pa����2���1��RA��p���\|�,�hence�����&�fe
�^�]ڍ�a���̽p������a���̽p����=��p�,�so����Жk�j�a���̽p���]�j�UR�=������=�p���
UT���=�z�孟Í�p����;:��%�������The�xcase�of��"��unramified�a��32t��p�.��7�So���here�w��re�can�talk�of��"�(�p�),��and�it�is�going�to����b�S�e�ʽa�ro�ot�of�1�in��E������2���t�.��It�is�supp�osed�to�remind�y��rou�of�the�T��Vate�curv�e�(elliptic�curv�e�with���m��rultiplicativ�e��reduction�at��p�).�8�No��rw��!n����������̽����Ͽj���̽D���p���)[�=���UR��f\� �����d���?�����̽����*:M���������J�0���'.�����̽������8F	��f\�!���B0��:��!�I��Here�b%����̽��ʬ�=�UR����̽�������̽����and�����̽���=�UR����̽���=��det���=��"���̽����,�}sso�b%plugging�in�w��re�get������2���O�2��RA������=�UR�"�.�_So�����̽����is�unramied,����ͩ����̽��uZ�(�frob������̽p��WO�)�UR=��a���̽p��GM����̽���(�frob������̽p��WO�)�=��pa���̽p���]�:����������̽���$¿j���̽D���p���lU�is��Lan�indecomp�S�osable�represen��rtation:��the����in�the�ab�o��rv�e��Lform�ula�is�non-zero.� lSince�����p�UR�6�=���,��this�prop�S�ert��ry�c�haracterizes�it.�8�W��Ve�can�sa�y�that�� ������������̽����y �=�UR� ���̽�� � �������f\� �����d����T� ���̽����$꨿�������P�0���$��1�����/꩟�f\�!���9�S�;�� ����where�߀the�second�term�in�the�tensor�pro�S�duct�is�the�unique�indecomp�osable���-adic�represen-���tation��of��G��.�al�C��(�����fe #�� n��Q���� #���T�p����=�Q���̽p���]�)�with�semisimpli cation�1������ ���̽����.�8�Corresp�S�ondingly���˿޼���̽p�� ��=�UR� �T�� ����st;����where�-�st��is�a�pun�for�b�S�oth�the�\standard�represen��rtation"�and�the�\Stein�b�S�erg�represen�tation".����Morally��V,��a���PS���represen��rtation�attac�hed�to�(1�;����jj�:�jj�)�is�not�irreducible,��but�it�has�a�unique���irreducible��quotien��rt.�������Remarks.�޹A�{��sp��ffe�cial��'r�epr�esentation�{��is�an��ry�ab�S�elian�c�haracter�tensored�with�the�Stein�b�S�erg���represen��rtation.������Langlands��'in��rv�en�ted�a�dictionary�b�S�et�w�een�represen�tations�of��GL��\�(2�;����Q���̽p���]�)�and�represen�ta-���tions���of��G��.�al�C��(�����fe #�� n��Q���� #���T�p����=�Q���̽p���]�),�ˁwhic��rh�go�S�es�through�complex�represen�tations�of�a�smaller�group�(the����Weil�35gr��ffoup�).��� ����62����?�A��K����b�{�K������{���0.1��,G�The�z�w��u�eigh�t�and�fundamen�tal�c�haracters��b#��Let����������UR�:��Gal���(�����fe #�� n��Q��� #��=�Q�)��!���GL����(2�;����F���̽���������r�)���d�b�S�e�Πan�irreducible�mo�dular�mo�d����represen��rtation.�/�Let��D���=�UR�D���̽p�� ��=��Gal���(�����fe #�� n��Q���� #���T�p����=�Q���̽p���]�)�Πb�e�the�decom-���p�S�osition��group�at��p�.�#�T��Vo�da��ry�w�e�will�see�ho�w�studying�the�lo�S�cal�prop�erties�of����help�us�to���understand��the�follo��rwing�theorem.���Ǎ���Theorem��27���L���If�О��UR>��2��and����is�mo��ffdular�then����is�mo�dular�of�a�c�ertain�weight��k�g �(��)��and�level����N�@�(��)�.����Since��˼��is�mo�S�dular,�����K������&�fet�]ڍ����� ȿ��T�;f�� Mw�for���some�mo�dular�form��f�G��.��HThe�theorem�asserts�that�w��re�can���actually�utak��re��f�y��2�1��S���߽k�6��(��)��]�(���̺1����(�N�@�(��))).��FAs�w�e�ha�v�e�seen�b�S�efore��N�@�(��)�is�the�prime�to����Artin���conductor��of���.������T��Vo�S�da��ry�'�w�e�attac�k�the�question�of�what��k�g �(��)�should�b�S�e.���T��Vo�do�this�w�e�restrict�to�the�inertia���group��2�I���̽p���]�.��Before�doing�this�w��re�in�tro�S�duce�a�certain�pair�of�c�haracters.��First�w�e�pro�v�e�a�lemma.���Let��ؼ���b�S�e�the�semisimpli cation�of���j���̽D�����.�pTh��rus����is�either�a�direct�sum�of�t��rw�o���c�haracters�or�����j���̽D�� �L�dep�S�ending��on�whether�or�not���j���̽D���is�irreducible.������Lemma��5���<uQ�The�35r��ffepr�esentation����n�is�tame,�i.e,����is�trivial�on�the���-Sylow�sub��ffgr�oup�of��I���̽�����.���Pr��ffo�of.�:6�Let����P�,��b�S�e�the���-Sylo��rw�subgroup�of��I���̽����.��Since��I���̽���۹is�normal�in��D�ރ�and��P��is�unique�it���follo��rws��that��P��n�is�normal�in��D�S��.�8�Let��W���=�UR�F���̽�������� G�����F���̽�������� ��b�e�the�represen��rtation�space�of���n9�.�8�Then������wF�W���Ɵ� ��P�� �=�UR�f�w����2��W���:���n9�(� �W �)�w��=��w�� ��for��all��4-2� ��o�2��P��ƿg����is��Ga�subspace�of��W�� �in��rv��X�arian�t��Gunder�the�action�of��D�S��.�^�T��Vo�see�this�let�� �2�2� ��D�Jչand�supp�ose�����w����2�UR�W���Ɵ��2�P�����.�8�Since�꨼P��n�is�normal�in��D�S��,�� ������2��1�� p � �W �h�=�� �����2�0����for��some�� ��W ���2�0��z��2��P��ƹ.�8�Therefore�����jǼ�n9�(� ���)��� ���1�� \|���(� �W �)���(� ��)�w����=�UR���(� ��W �� ��0��%V�)�w��=�UR�w����so�꨼�n9�(� �W �)���(� ���)�w����=�UR���(� ��)�w�=R�hence�꨼��(� ��)�w����2�UR�W���Ɵ��2�P�����.������But�� �W���Ɵ��2�P�� �6�=�UR0.�'�T��Vo�see�this�write��W�X�as�a�disjoin��rt�union�of�its�orbits�under�the�action�of��P��ƹ.���Since��-�P�4�is�an���-Sylo��rw�group�and��W��is� nite�w��re�see�that�the�size�of�eac�h�orbit�is�either�1�or�a���p�S�ositiv��re�O�p�o�w�er�O�of���.�;No�w��f�0�g��is�a�singleton�orbit,�n��W��has���-p�S�o�w�er�order,�n�and�all�non-singleton���orbits��ha��rv�e�order�a�p�S�ositiv�e�p�S�o�w�er�of����so�there�m�ust�b�S�e�at�least���.���1�other�singleton�orbits.���Eac��rh��of�these�other�singleton�orbits�giv�es�a�nonzero�elemen�t�of��W���Ɵ��2�P�����.������If��.�W���Ɵ��2�P���̹=�7:�W�=�then��P��acts�trivially�so�w��re�are�done.� MqIf��W���Ɵ��2�P���̿6�=�7:�W��then��W���Ɵ��2�P��O��is�a�one���dimensional��subspace�in��rv��X�arian�t��under��D��so�b��ry�semisimplicit�y��� =�is�a�diagonal�represen�tation.���Supp�S�ose�꨼ ��o�2�UR�P��n�then�� �AŹhas�order������2�n�� ���for�some��n�.�8�W��Vrite�� ���Iܼ�n9�(� �W �)�UR=�����f\� �����d���?�� ���"���0�������d0���!�Ƽ �����.3��f\�!����"*���then�꨼ ������2����-:�n��� ��=�UR1�and�� ���O���2����-:�n��� ��=�1.�8�Since�� ��;��� ����2��F���̽�������� ��it�follo��rws�that�� �h�=�� ����=�1.�h)>���� �����63����@���K����b�{�K������{����The�ܒlemma�implies���n9�j���̽I���߹factors�through�the�tame�quotien��rt��I���̽t��� �=�UR�I��=P��ƹ.�4.W��Ve�no�w�describ�S�e��I���̽t������explicitly��V.� RLet�d��Q����2��tame��RA�p����˹b�S�e�the�maximal�tame�extension�of��Q���̽p��,[�and�let��K�1�=�UR�Q����2��ur��RA�p��� �>�b�e�the�maximal���unrami ed��extension.�8�Since��P��n�is�the�part�of�inertia� xing��Q����2��tame��RA�p����͹,��ߍ�� �I���̽t��� �=�UR�I��=P���=��Gal���(�Q��� ���tame���ڍp����ͼ=�Q��� ���ur���ڍp��� [email protected]�)�:����F��Vor��eac��rh��n��prime�to��p��there�is�a�to�w�er�of� elds��E�����Ӆ ������Q����2��tame��RA�p����������5��j���ꕍ���+�K�ܞ�(�p����L䍑���1��33�x|�fe�7�����n�������)��������5��j������Ӻ̼K�1�=�UR�Q����2��ur��RA�p���������5��j�������k/�Q���̽p������H�/��By�m�Kummer�theory��Gal���n(�K�ܞ�(�p����L䍑���1��33�x|�fe�7�����n�������)�=K��)�UR=�����̽n���P�(�K�ܞ�)�m�where�����̽n���(�K�ܞ�)�denotes�the�group�of��n�th�ro�S�ots�of����unit��ry�*�in��K�ܞ�.���Th�us�for�eac�h��n��prime�to��p��w�e�obtain�b�y�restriction�a�map��I���̽t��T��!�¶����̽n���P�(�K�ܞ�).���They���are��compatible�so�passing�to�the�limit�w��re�obtain�a�map��ߍ�x��I���̽t��� �!���UR�lim������ �n���� ����̽n���P�(�K�ܞ�)�UR=����lߟ����Y��� 8獓�r�<r�6�=�p�����K��lim�����z� �a���&�A����̽r��<r����a��� �(�K��)�=����lߟ����Y��� 8獓�r�<r�6�=�p���K��Z���̽r���b�(1)�:��#�S��View��red��more�clev�erly�mo�S�d��p��w�e�obtain�a�map������ʼI���̽t��� �!���UR�lim������ �n���� ����̽n���P�(�����fe|r� n��F����|r��̽p�� CϹ)�UR=���lim������ �n����F��� ������ڍ�p�����n���� 唼:�� ?�����W��Ve�(�no��rw� x�atten�tion�on��n�UR�=��p����2�i���p�� ��1.��=The�(�p�S�oin�t�is,�O�that�for�ev�ery�suc�h��i��w�e�ha�v�e�a�standard����map�����Q1�I���̽t��� �!�UR�F��� ������ڍ�p�����i�����ϼ:��ԩ��W��Ve��Ncall�this�map�the��fundamental��char��ffacter�of�level��i�.���This��Nis�unnatural�b�S�ecause�w��re�had���to��,tak��re�an�em�b�S�edding��F���̽�������� xm�,���!��������fe|r� n��F���� Xm��̽��QS�.��kInstead�Serre�b�egins�with�some�disem��rb�o�died� eld��F�q�of���order�A¼p����2�i��dڹ.� >/There�are��i��di eren��rt�maps��F�� ��p�����i��� �!�!��R�F�㈹corresp�S�onding�to�the��i��automorphisms�of����F�� ��p�����i����Ϲ.�,DRestricting���these�maps�to��F����2�����V��p�����i���� ���and�comp�S�osing�with�the�fundamen��rtal�c�haracter��I���̽t��� �!�UR�F����2�����V��p�����i�������de ned�� ab�S�o��rv�e�giv�es�Serre's�fundamen�tal�c�haracters�of��I���̽t���Թwith�v��X�alues�in��F��ƹ.�jW��Ve�ha�v�e�th�us���de ned��some�c��rharacters�on��I���̽t��|r�whic�h�can�b�S�e�view�ed�as�standard.������Recall��that�w��re�ha�v�e�a�mo�S�d������2��� :�represen�tation����whic�h�giv�es�rise�to��ߍ�� ���Ë�:�UR�I���̽t��� �!���GL����(2�;����F���̽���������r�)�:����Since���the�elemen��rts�of��I���̽t��IĹha�v�e�order�prime�to�the�c�haracteristic����this�represen�tation�is�semisim-����ple�52so�it�can�b�S�e�diagonalized�up�on�passing�to�an�algebraic�closure�of��F���̽���������r�.��cTh��rus����k�corresp�onds���to��a�pair�of�c��rharacters�����"� ��;��� ����:�UR�I���̽t��� �!��F��� ������ڍ������2����� G�:��� �����64����A����K����b�{�K������{����These��c��rharacters�ha�v�e�some�stabilit�y�prop�S�erties�since���q׹is�the�restriction�of�a�homomor-����phism��from�the�full�decomp�S�osition�group.�8�Consider�the�to��rw�er��of� elds�������ҼK�ܞ�(�p�����O�����1��33�x|�fe�7�����n�������)�UR���K�1�=��Q��� ���ur���ڍp��� Ӓ���Q���̽p���]�:�� ���Let�X �G�UR�=��Gal���(�K�ܞ�(�p����L䍑���1��33�x|�fe�7�����n�������)�=�Q���̽p���]�).�Recall�that��Gal����(�K��(�p����L䍑���1��33�x|�fe�7�����n�������)�=K��)�UR=�����̽n���P�(�K�ܞ�)�and��Gal����(�K�5�=�Q���̽p���]�)�is�generated���b��ry��o{F��Vrob��� ���̽p�� ��.��ZWhat�o{happ�S�ens�is�that�if��h�7h�2�����̽n���P�(�K�ܞ�)�o{and��g����2�7h�G��is�suc�h�that��g�ݴ�restricts�to��F��Vrob��� ���̽p�����then�u�w��re�ha�v�e�a�conjugation�form�ula:��y�g�n9hg�����2��1�� �=�UR�h����2�p���]�.��Applying�u�this�reasoning�to�the�����situation���with�꨼h�UR�2��I���̽t��|r�and��g�X�restricting�to��F��Vrob����ן�̽p��$Eܹw��re�nd�that�����h��n9�(�g�hg�������1��ʵ�)�UR=���n9�(�h�����p���]�)�=����(�h�)�����p���]�:����Th��rus��Othe�pair�of�c�haracters��f���;�����O�g��is�stable�under��p�th�p�S�o�w�ering,���i.e.,��f���;�����O�g�UR�=��f������2�p����;������O���2�p��q��g�.� mThe���p�S�oin��rt��is�that�����ψ��n9�(�g��)���(�h�)���(�g�������1��ʵ�)�UR=���n9�(�g�hg�������1���)�UR=���n9�(�h�)�����p���]�:����Th��rus�6the�represen�tation��h����7!���n9�(�h�)����2�p��	�z�is�6isomorphic�to��h����7!����(�h�)�6via�conjugation�b��ry����(�g��).���Th��rus�n׼���,����&�as�a�pair�are�the�same�as�������2�p����,������O���2�p��
�as�a�pair�since�they�came�from�isomorphic���represen��rtations.������What���do�S�es�this�mean?�2�One�p�ossibilit��ry�is�that���h�=�UR�������2�p�����and������=�����O���2�p��q��.�2�This�means����;������tak��re���v��X�alues�/in��F����2����RA��p����]�.�dThe�other�p�S�ossibilit��ry�is�that��������2�p���j�=��~���ιand�����O���2�p��	<*�=�����.�dThen�������2�p���-:�2�����=��~��C�and�����O���2�p���-:�2���
fѹ=������so�Q����;������tak��re�v��X�alues�in��F����2��������p�����2������.�nThe�rst�situation�in�whic�h���eD�and�����tak�e�v��X�alues�in��F����2����RA��p���	�is�called���the���level�351��situation.�8�The�second�situation�is�called�the��level�2��situation.������W��Ve���will�pla��ry�a�carniv��X�al�game,��z\guess�y�our�w�eigh�t."�lrFirst�w�e�consider�the�lev�el�2�case.���Our���strategy�is�to�try�and�express����x�and���+8�in�terms�of�the�t��rw�o���fundamen�tal�c�haracters�of���lev��rel��2.������First��2some�bac��rkground.�/}Before�stating�the�general�result�w�e�talk�ab�S�out�a�sp�ecial�case.���Supp�S�ose�2ѼE���is�an�elliptic�curv��re�o�v�er��Q��and��p��is�a�prime�(2�is�allo�w�ed).�[Assume��E���has�go�S�o�d���sup�S�ersingular��reduction�at��p�.�8�Then�there�is�a�represen��rtation������H�Gal����s(�����fe
#��	n��Q���
#��=�Q�)�UR�!���Aut��=(�E���[�p�])���whic��rh���ma�y�or�ma�y�not�b�S�e�irreducible.�gIt�giv�es�rise�via�restriction�to�t�w�o�c�haracters����;����<N�:����I���̽t��Tڿ!���F����2����RA��p����]�.��Serre���[�Gr��ffoup�es���de�Galois�attach��3��L�es�aux�p��ffoints�d'or�dr�e�ni�des�c�ourb�es�el���liptiques���sur��un�c��fforps�de�nombr�es�,���1972]���pro��rv�es�that����;����v�are�the�t�w�o�fundamen�tal�c�haracters�of�lev�el���1��and�that��I���̽t��|r�acts�tamely�and�irreducibly��V.�8�He�also�obtained�a�map�����f��I���̽t����!�UR�F���������ڍ�p�����2����GV����GL����(2�;����F���̽p���]�)���where����F����2��������p�����2�������sits�inside��GL����(2�;����F���̽p���]�)�via�the�action�of�the�m��rultiplicativ�e���group�of�a�eld�on�itself���after�!�c��rhoice�of�a�basis.���The�action�of��I���̽t���Źon��F����2��������p�����2����
1992].�Supp�S�ose����f����is�a�newform�of�w��reigh�t����k���suc�h�that�2�UR���k��o���p��and�that�the����lev��rel�@�of��f����is�prime�to��p��ȹ=���.�;T��Vak�e��@����&�fet�]ڍ�����
P(��T�f�h�;����where�@��E���̽f��	^�=��Q�(�co�S�ecien�ts��of��H�h�f�G��)�@�and����is�a�prime�of����O���̽E��i?�f���X��dividing��M�p�.�1Supp�S�ose�w��re�are�in�the�sup�ersingular�case�so��a���̽p�����UR�0��mo�d��X���.�1Semisimplifying���and��restricting�as�b�S�efore�giv��res�a�pair�of�c�haracters����;�������:�UR�I���̽t����!��F����2����RA��p���������	j�.��hv����Theorem��28���L���Under���the�ab��ffove�hyp�othesis,��˿f���;�����O�g��e�quals��f� ��n9���2�k�6���1��nG�;����(� ��n9���2�0��<r�)����2�k�6���1���g��wher�e�� �/)�and�� ��n9���2�0���b�ar�e���the�35two�fundamental�char��ffacters�of�level��2�,�i.e.,�� �Ë�=�UR(� ��n9���2�0��<r�)����2�p�����and�� ��n9���2�0���Ĺ=�� ��n9���2�p��5��.������Edixho��rv�en's��{metho�S�d�is�to�reduce�from�w��reigh�t��{�k�5��to�w��reigh�t��{2.�/|Giv�en�a�mo�S�dular�form�mo�d����p�T��on����̺1����(�N�@�)�of�w��reigh�t�T��k����and�c��rharacter��"��he�sho�ws�that�it�is�enough�to�lo�S�ok�at�a�corresp�onding���mo�S�dular�Y}form�on����̺1����(�N�@�p�)�of�w��reigh�t�Y}2�and�c��rharacter��"!��n9���2�k�6���2���Ĺwhere��!�Ƕ�is�a�T��Veic�hm�uller�c�haracter���on���(�Z�=p�Z�)����2�����.�~This�is�a�metho�S�d�whic��rh�helps�us�understand�what�happ�ens�at�primes��p��where�a���certain��fab�S�elian�v��X�ariet��ry�do�es�not�ha��rv�e��fgo�o�d�reduction�but�has�p�oten��rtially�go�o�d�reduction.�(�In���this���direction�Ogus�suggests�reading�the�pap�S�er�[F��Valtings�and�Jordan,����Crystal���line�Ѩc��ffohomolo�gy���and��35�GL����(2�;����Q�),��Israel�J.�Math.�8�90�(1995),�no.�1-3,�1{66.]������What�ܽcan�w��re�guess�ab�S�out�the�arbitrary�situation?�4<The�situation�is�as�follo�ws.�4<W��Ve�b�S�egin���with��Ga�represen��rtation����whic�h�giv�es�rise�to�a�represen�tation���\��whic�h�in�turn�giv�es�rise�to���a�դpair�of�fundamen��rtal�c�haracters�� ��n9���2�a��
G��and�(� ��n9���2�0��<r�)����2�a��	��=��J� ��n9���2�pa��	�e�.���(Here��a��should�b�S�e�though�t�of�as�a���n��rum�b�S�er��mo�d��p����2�2��p׿��ӹ1.)�TThe�condition�that�w��re�are�not�in�lev�el�1�means�that��a��is�not�divisible���b��ry�꨼p����+�1�since�����WX� �n9 ������0���Ĺ=�UR� ������p�+1��gd�:��I���̽t����!��F���������ڍ�p�����;��is��the�unique�fundamen��rtal�c�haracter�of�lev�el�1�(namely�the�mo�S�d��p��cyclotomic�c�haracter���).������W��Ve��7next�do�a�Euclidean�division.�7�T�ak��re�0�O���a�<�p����2�2�����X��1��7and�write��a�O�=��q�n9p��+��r�S��.�7�Then����pa��e�=��q�n9p����2�2���M�+��I�r�S�p�,�Bbut�0�w��re�are�w�orking�mo�S�d��p����2�2���M���I�1�so�this�b�ecomes��pa��e�=��q�H��+��I�r�p�.�
�What�0�are�the���p�S�ossible�(�v��X�alues�for��q���and��r��?��^By�the�Euclidean�algorithm�0��"���r�����p������1�(�and�0��"���q�-[���p������1.���No��rw���what�are�the�constrain�ts�on��r�-�and��q�+عtogether?�)�The�main�constrain�t�is�that��r���6�=�UR�q�n9�,�Ơsince���if�˼r��7�=����q���then��a��is�a�m��rultiple�of��p��˹+�1.��ISo��w�e�can��assume��that�0������r��7<�q����p��˿��1.��IW��Ve��kno�w���that��������f���;�����O�g�UR�=��f�(� �n9 ������0��<r�)�����r���b�(� ������0���)�����q�I{��r��Cؼ;����(� � ������0���)�����r���b� ������q�I{��r����g�:����F��Vor��example,�if���h�=�UR� ��n9���2�a��	\��then��d����^��h�=�UR� ��n9����q�I{p�+�r��N��=�(� ��n9����0��<r�)�����q����� ��n9����r��U�=�(� �n9 ������0���)�����r���b�(� ������0���)�����q�I{��r��Cؼ:����Since�6¼ �n9 �����2�0��L�=��ڼ��w��re�can�view��f���;�����O�g��as�a�pair�of�c�haracters�(� ��n9���2�0��<r�)����2�q�I{��r��z��and�� ��n9���2�q�I{��r���ӹwhic�h�has�b�S�een����m��rultiplied��as�a�pair�b�y������2�r���b�.������What��w��rould�happ�S�en�if�for�example��r���=�UR0?�8�In�this�case��d����>��f���;�����O�g�UR�=��f�(� ��n9����0��<r�)�����k�6���1���;� ��n9����k�6���1��nG�g����where�꨼k��o�=�UR�q��+���1.�8�So�when��r���=�0�w��re�guess�the�w�eigh�t�to�b�S�e��k�g�(��)�UR=��q��+���1.�������What��9happ�S�ens�more�generally?��Supp�ose��f��8�is�a�mo�dular�form�though��rt�of�mo�d��p��where�����j�p�.�}�If�V�f�U��=��
������P�����a���̽n���P�q��n9���2�n��
���.�o�In���[Lecture�Notes�in�Math,��no.�601]�w��re�nd�that���S�f��q�=��tr�����P����na���̽n���P�q��n9���2�n��
z�is�a�form�giving�rise�������66����C���K����b�{�K������{��to����,D�
���.�s�If��k�"�is�the�w��reigh�t�of��f�G��,�؜then���S�f���has�w�eigh�t��k��a�+�,D�p��+�1.�s�More�generally��V,�؜since������2�r������app�S�ears��in�the�pair��������f� ��n9����q�I{��r����;����(� ��n9����0��<r�)�����q�I{��r��Cؿg����������r�����w��re��obtain�������k�g�(��)�UR=��q������r��6�+�1�+�(�p��+�1)�r���=�UR�q��+��pr��6�+�1�:������But��Xb�S�e�careful,��the�w��reigh�t��X�k�0u�do�es�not�ha��rv�e��Xto�go�up.�-�In�fact,��it�go�es�up��p�f����1��Xtimes�then���ends�S�up�where�it�started.�s�This�giv��res�rise�to�the�theory�of�theta�cycles�whic�h�w�e�will�study���in��the�next�lecture.�������67���������;��K���GD�:F
C�
cmbxti10�9�-�
cmcsc10�3���@cmti12�2��N�ffcmbx12�0\��%eufm8�/�%n�
eufm10�-2�@�cmbx8�,��N�cmbx12�)���
msbm10�&����
msam10�%߆�Tcmtt12�\$��N�G�cmbx12���g�ffcmmi12�X�Qffcmr12�D��t�qG�cmr17�q�%cmsy6��K�cmsy8�!",�
cmsy10�;�cmmi6��2cmmi8���g�cmmi12��Aa�cmr6�|{Ycmr8�X�Qcmr12�
�b>

cmmi10�K�y

cmr10��O�

line10���u

cmex10�������