Sharedwww / job / Abstracts.dviOpen in CoCalc
����;� TeX output 1999.11.12:1118������ǂ����6�;��䃍�������������	�F
C��q
cmbxti10�Wil�8�liam��sA.�Stein��іA��c�bstr���Uacts��(�����d��k�ff���������2ō��|4V�������N��qcmbx12�Publications�����f�R�X�Qcmr12�(published,��accepted�for�publication,�or�submitted)��������[�1.����_P���N�cmbx12�Lectures�W�on�mo�`dular�forms�and�Galois�represen��tations�謹(170�pages),�Ewith�K.�Ri-����_Pb�S�et:��oIn�G�1996,��BKen�Rib�et�taugh��rt�an�adv��X�anced�course�on�mo�dular�forms�and�Galois����_Prepresen��rtations.�	@In�B_collab�S�oration�with�Rib�et,��MI�Bam�turning�m��ry�course�notes�in�to�a����_Pb�S�o�ok�o�that�has�b�S�een�accepted�for�publication�in�Springer-V��Verlag's�Univ��rersitext�series.����_PThe�M�c��rhapters�are:���Ov�erview;��4Mo�S�dular�represen�tations;��4Mo�S�dular�forms;�Em��rb�S�edding����_PHec��rk�e���op�S�erators�in�the�dual;�i�Rationalit��ry�and�in�tegralit�y�questions;�i�Mo�S�dular�curv�es;����_PHigher�i�w��reigh�t�mo�S�dular�forms;���Newforms;�Some�explicit�gen��rus�computations;�The�eld����_Pof���mo�S�duli;��EHec��rk�e�op�S�erators�as�corresp�ondences;��EAb�elian�v��X�arieties�from�mo�dular�forms;����_PThe�M�Gorenstein�prop�S�ert��ry;��2Lo�cal�M�prop�erties�of����g�cmmi12�������2cmmi8���uZ�;��2Serre's�conjecture;�F��Vermat's�Last����_PTheorem;�
�Deformations;�The���Hec��rk�e�algebra��T�����|{Ycmr8������.�yTThough�a�v�ersion�of�the�notes�has����_Pb�S�een��circulating�for�a�few�y��rears,�signican�t�p�S�olishing�remains�to�b�e�done.��������[�2.����_P�HECKE:���The�mo�`dular�forms�calculator:��ҹI�xowrote�x���-�
cmcsc10�Hecke�,��^a�p�S�opular��߆�Tcmtt12�C++��pac��rk��X�age����_Pfor���computing�with�spaces�of�mo�S�dular�forms�and�mo�dular�ab�elian�v��X�arieties.�rEI���ha��rv�e����_Pb�S�een���in��rvited�to�visit�the��Ma���gma��group�in�Sydney�in�order�to�mak�e��Hecke��a�part�of����_Ptheir��computer�algebra�system,�and�I�ha��rv�e��already�p�S�orted�m��ry�co�de�to��Ma���gma�.�������[�3.����_P�Explicit�m�approac��hes�to�mo�`dular�ab�elian�v��@arieties�ڟ�(130�pages),���Ph.D.�thesis�un-����_Pder�V�H.��WW.�Lenstra:��In�the�rst�part�of�m��ry�thesis,���I�VXgiv�e�algorithms�for�computing����_Pwith�>�mo�S�dular�ab�elian�v��X�arieties.�5rThese�include�algorithms�for�computing�congruences����_Pb�S�et��rw�een�\�mo�dular�eigenforms,�ythe�mo�dular�degree,�ythe�rational�parts�of�the�sp�ecial�v��X�al-����_Pues���of�the��L�-function,��1comp�S�onen��rt�groups�at�primes�of�m�ultiplicativ�e�reduction,��1p�S�erio�d����_Plattices,��and�the�real�v��rolume.�����_PIn�:�the�second�part�of�m��ry�thesis,���I�:`use�these�algorithms�to�in�v�estigate�sev�eral�op�S�en����_Pproblems.�The��
rst�concerns�the�Birc��rh�and�Swinnerton-Dy�er�conjecture,��whic�h�ties����_Ptogether�uthe�constellation�of�in��rv��X�arian�ts�uattac�hed�to�an�ab�S�elian�v��X�ariet�y;���I�'v�erify�this����_Pconjecture�YLfor�certain�sp�S�ecic�ab�elian�v��X�arieties�of�dimension�greater�than�one.�lThe�k��rey����_Pidea��is�to�use�B.�Mazur's�notion�of�visibilit��ry��V,���coupled�with�explicit�computations,�to����_Ppro�S�duce���lo��rw�er�b�S�ounds�on�Shafarevic�h-T��Vate�groups.��*I��3also�describ�S�e�related�computations����_Pthat�C>are�used�in�an�up�S�coming�join��rt�pap�er�with�L.�Merel�on�rational�p�oin��rts�of��X���̽0����(�N�@�)����_Po��rv�er�꨼Q�(������p���]�).����_PE.�[�Artin�conjectured�in�1924�that�the��L�-series�asso�S�ciated�to�a�represen��rtation���UR�:��G�����2�@�cmbx8�Q��1��!",�
cmsy10�!�����_P�GL���-���̽2��2��(�C�)���is�en��rtire.�e�K.�Buzzard�and�I��dv�erify�this�conjecture�in�eigh�t�new�icosahedral����_Pcases;�<]the��8argumen��rt�uses�a�theorem�of�K.�Buzzard�and�R.�T��Va�ylor�along�with�a�computer����_Pcomputation,��whic��rh�is�used�to�ll�in�a�missing�mo�S�d-5�mo�dularit��ry�result.����_PFinally��>I��)turn�to�Serre's�conjecture.�I�consider�a�family�of�exceptional�cases,���and�sho��rw����_Pthat�]the�conjecture�app�S�ears�to�fail�as�badly�as�p�ossible.�	�*I�\�also�describ�e�n��rumerical����_Pexp�S�erimen��rts��to�w�ards�a�mo�S�d-�pq�X�extension�of�Serre's�conjecture.����_PI��exp�S�ect�to�nish�m��ry�thesis�b�efore�Ma��ry�2000.������*�ǂ����6�;��䃍�������������	�Wil�8�liam��sA.�Stein��іA��c�bstr���Uacts��(�����d��k�ff���������2ō��y�;�����[̹4.����_P�Lectures���on�Serre's�conjectures��(77�pages),�}with�K.�Rib�S�et:��iThis�is�an�exp�ository�����_Ppap�S�er��based�on�Ken�Rib�et's�lectures�at�the�1999�P��rark�Cit�y�Mathematics�Institute;�4Rit����_Pwill�Lb�S�e�published�in�the�IAS/P��rark�Cit�y�Mathematics�Institute�Lecture�Series.�\�There����_Pare�:-9�lectures:���In��rtro�S�duction�to�Serre's�conjectures;�a�The�w�eak�and�strong�conjectures;����_PThe���w��reigh�t�in�Serre's�conjecture;�;}Galois�represen�tations�from�mo�S�dular�forms;�;}In�tro�S�duc-����_Ption���to�lev��rel�lo�w�ering;���Approac�hes�to�lev�el�lo�w�ering;���Mazur's�principle;�Lev��rel�lo�w�ering����_Pwithout�r�m��rultiplicit�y�one;��Lev�el�lo�w�ering�with�m�ultiplicit�y�one;��Other�directions.�ѴW��Ve����_Pha��rv�e�P�nished�all�but�the�last�three�sections,�jand�exp�S�ect�to�nish�b��ry�early�Decem�b�S�er,����_P1999.���������[�5.����_P�Empirical��,evidence�for�the�Birc��h�and�Swinnerton-Dy�er�conjecture�for�mo�`d-����_Pular�z�Jacobians�of�gen��us�2�curv�es����(22�pages),��with�E.��WV.�Flynn,�F.�Lepr��r��s�ev�ost,����_PE.��WF.�RSc��rhaefer,�p�M.�Stoll,�J.��WL.�W��Vetherell:��W�e�Rpro��rvide�systematic�n�umerical�evidence�for����_Pthe��2BSD���conjecture�in�the�case�of�dimension�t��rw�o.��}This��2conjecture�relates�six�quan-����_Ptities�{nasso�S�ciated�to�a�Jacobian�o��rv�er�{nthe�rational�n��rum�b�ers.��1One�{nof�these�quan��rtities�is����_Pthe�:size��S���of�the�Shafarevic��rh-T��Vate�group.���Unable�to�compute��S��directly��V,�"^w��re�compute����_Pthe��qv��re�other�quan�tities�and�solv�e�for�the�conjectural�v��X�alue��S���̽?��z�of��S��׹.�\:F��Vor�all�32�curv�es����_Pconsidered,�~mthe�`�real�n��rum�b�S�er�`߿S���̽?���q�is�v��rery�close�to�either�1,�2,�or�4,�and�agrees�with�the����_Psize��of�the�2-torsion�of�the�Shafarevic��rh-T��Vate�group,�S�whic�h�w�e�could�compute.���This����_Ppap�S�er��has�b�een�submitted.�������[�6.����_P�P��arit�y���structures�and�generating�functions�from�Bo�`olean�rings���(8�$�pages),����_Pwith�m�D.�Moulton:��jLet��S� ��b�S�e�a�nite�set�and��T���b�e�a�subset�of�the�p�o��rw�er�m�set�of��S��׹.�<Call��T����_P�a�gG� ���@cmti12�p��ffarity��rstructur�e��for��S��if,���for�eac��rh�subset��b��of��S��of�o�S�dd�size,���the�n��rum�b�S�er�of�subsets�of��b����_P�that�@Hlie�in��T���is�ev��ren.�9�W��Ve�classify�parit�y�structures�using�generating�functions�from�a����_Pfree���b�S�o�olean�ring.�'�W��Ve�also�sho��rw�that�if��T�Y��is�a�parit�y�structure,���then,�for�eac��rh�subset��b����_P�of���S�k��of�ev��ren�size,���the�n�um�b�S�er�of�subsets�of��b��of�o�dd�size�that�lie�in��T�Z��is�ev��ren.�(JW��Ve�then����_Pgiv��re��sev�eral�other�prop�S�erties�of�parit�y�structures�and�discuss�a�generalization.�This����_Ppap�S�er��has�b�een�submitted.�������[�7.����_P�F���allacies,��iFla��ws,�and�u!Flim
am�#92:�	�BAn�Inductiv�e�F���allacy�,��(1���page)�with����_PA.��Riskin:�8�College�Math.�Journal�26:5�(1995),�382.��1.����u�W��=ork��
in�progress�������[̹1.����_P�The���mo�`dular�forms�database:��ùThis��is�an�ev��rolving�collection�of�tables�of�mo�S�dular����_Peigenforms,��sp�S�ecial���v��X�alues�of��L�-functions,�arithmetic�in��rv��X�arian�ts���of�mo�S�dular�ab�elian����_Pv��X�arieties,��and�other�data.�8�These�tables�are�a��rv�ailable�at����_P�http://shimura.math.berkeley.edu/~was/Tables/�.�������[�2.����_P�Visibilit��y��'of�Shafarevic�h-T���ate�groups�of�mo�`dular�ab�elian�v��@arieties�?T�(20�pages),����_Pwith���A.�Agash��r��s�e:��W��Ve�study�Mazur's�notion�of�visibilit�y�of�Shafarevic�h-T��Vate�of�mo�S�dular����_Pab�S�elian���v��X�arieties,��xand�use�it�to�v��rerify�the�conjecture�of�Birc�h�and�Swinnerton-Dy�er�for����_Psev��reral�
�sp�S�ecic�ab�elian�v��X�arieties.��qW��Ve�also�giv��re�evidence�that�visibilit�y�is�rare.��qThe����_Pcomputations��&are�done,��@but�w��re�are�still�adding�to�the�theoretical�con�ten�t�of�the�pap�S�er����_Pand��p�S�olishing�the�presen��rtation.������d�ǂ����6�;��䃍�������������	�Wil�8�liam��sA.�Stein��іA��c�bstr���Uacts��(�����d��k�ff���������2ō��y�;�����[̹3.����_P�Comp�`onen��t��Tgroups�of�optimal�quotien�ts�of�Jacobians���(16�pages):��	Let��A��b�S�e�����_Pan���optimal�quotien��rt�of��J���̽0����(�N�@�).��WThe�main�theorem�of�this�pap�S�er�giv�es�a�relationship����_Pb�S�et��rw�een�;Wthe�mo�dular�degree�of��A��and�the�order�of�the�comp�onen��rt�group�of��A�.�*�F��Vrom����_Pthis��GI��Ededuce�a�computable�form��rula�for�the�comp�S�onen�t�group�of�an�y�optimal�quotien�t����_Pof����J���̽0����(�N�@�)�at�a�prime�of�m��rultiplicativ�e���reduction.���I���then�compute�o��rv�er���one�thousand����_Pexamples�64leading�me�to�conjecture�that�the�torsion�and�comp�S�onen��rt�groups�of�quotien�ts����_Pof�I��J���̽0����(�p�),��Kfor��p��prime,�are�as�simple�as�p�S�ossible.�	U�This�pap�er�is�almost�ready�to�b�e����_Psubmitted.��������[�4.����_P�Mo�`d��׹5��approac��hes�to�mo�dularit��y�of�icosahedral�Galois�represen�tations�Vq�(16����_Ppages),��ywith�2�K.�Buzzard:�ȕConsider�a�con��rtin�uous�2�o�S�dd�irreducible�represen��rtation����^�:�����_PGal��/�(�����fe
#��	n덼Q���
#��=�Q�)���!���GL����~��̽2��U��(�C�)�:�1�A�~�sp�S�ecial�case�of�a�general�conjecture�of�Artin�is�that�the����_P�L�-function��ʿL�(�;���s�)�asso�S�ciated�to����is�en��rtire.�
�FBuzzard�and�I��Qgiv�e�new�examples�of����_Prepresen��rtations�׿��that�satisfy�this�conjecture.��W��Ve�obtained�these�examples�b�y�applying����_Pa���recen��rt�theorem�of�Buzzard�and�T��Va�ylor�to�a�mo�S�d�5�reduction�of���,��com�bined�with����_Pa�0jcomputational�v��rerication�of�mo�S�dularit�y�of�a�related�mo�S�d�5�represen�tation.�	
&The����_Pcomputations�rof�the�pap�S�er�are�complete,��:and�ha��rv�e�rb�een�mostly�written�up.��W��Ve�exp�ect����_Pto��nish�the�pap�S�er�in�time�for�the�Hot�T��Vopics�w��rorkshop�at�MSRI�in�Decem�b�S�er,�1999.�������[�5.����_P�Indexes�3-of�gen��us�one�curv�es����(10�pages):�p�I��spro��rv�e���results�ab�S�out�the�complexit��ry�of����_Pgen��rus�p�one�curv�es.��?The�greatest�common�divisor�of�the�degrees�of�the�elds�in�whic�h����_Pa�O,curv��re�of�gen�us��g��e�o�v�er��Q��has�a�rational�p�S�oin�t�divides�2�g�]P����2;��mwhen��g�n��=�h1�this�is�no����_Pcondition���at�all.�\�In�the�1950s�S.�Lang�and�J.�T��Vate�ask��red�whether,�9�giv�en�a�p�S�ositiv�e����_Pin��rteger��9�m�,���there�exists�a�gen�us�one�curv�e�so�that�the�smallest�n�um�b�S�er�eld�in�whic�h�it����_Phas���a�rational�p�S�oin��rt�is�of�degree��m��o�v�er��Q�.�1�Using�Euler�systems,���I���pro�v�e�this�when��m����_P�is��not�divisible�b��ry�4.�8�This�pap�S�er�is�almost�ready�to�b�e�submitted.������"'���;��ܙ��	� ���@cmti12�F
C��q
cmbxti10�߆�Tcmtt12��-�
cmcsc10�2�@�cmbx8�!",�
cmsy10��2cmmi8���g�cmmi12�|{Ycmr8���N�cmbx12���N��qcmbx12�X�Qcmr12�-(�������