Sharedwww / nsf / project_summary.dviOpen in CoCalc
����;� TeX output 2004.03.31:1814��������������������vI�����5��#�K�F
C�G�
cmbxti10�Wil��liam�)ZA.�Stein�Z�Pr���oje�ct�)ZSummary��vˎ�����'�8�m#�R

cmss10�(617)�UU308-0144�'OE�m#�R
�3
cmss10�w��[email protected]�rva�rd.edu�http://mo�M�dula�r.fas.ha�rva�rd.edu�����'�8��ff��������V��������'�8�K�`y
�3
cmr10�The�.pBirc��!h�and�Swinnerton-Dy�er�conjecture�and�Mazur's�notion�of�visibilit�y�of��
����'�8Shafarevic��!h-T��eate���groups�motiv��dDate�the�computational�and�theoretical�goals�of�this����'�8prop�M�osal.�a�The��Scomputational�goals�are�to�dev��!elop�new�algorithms,�Ntables,�and����'�8soft��!w�are�Y4for�studying�mo�M�dular�forms�and�mo�dular�ab�elian�v��dDarieties.��"The�theoreti-����'�8cal��/goals�are�to�pro��!v�e��/new�theorems�that�relate�Mordell-W��eeil�and�Shafarevic��!h-T�ate����'�8groups��fof�elliptic�curv��!es�and�ab�M�elian�v��dDarieties.��v��'�8��"V
�3
cmbx10�In��ttellectual�2�Merit:����'�8�The���PI���is�one�of�the�more�sough��!t-after�p�M�eople�b�y�the�w�orldwide�comm�unit�y�of�n�um-����'�8b�M�er���theorists,��for�computational�conrmation�of�conjectures,�for�mo�M�dular�forms����'�8algorithms,���for�n�data,�and�for�w��!a�ys�of�form�ulating�problems�so�as�to�mak�e�them����'�8more�Z
accessible�to�algorithms.��iThis�pro���ject�will�pro��!vide�new�computational�to�M�ols,����'�8including�\ha�ma���jor�new�pac��!k��dDage�for�computing�with�mo�M�dular�ab�elian�v��dDarieties�o��!v�er����'�8n��!um�b�M�er��elds,��and�enhance�the�mo�dular�forms�database,��whic��!h�is�used�b�y�man�y����'�8mathematicians��fwho�study�mo�M�dular�forms.��P���'�8The�Y�PI's�in��!v�estigations�Y�in�to�Mazur's�notion�of�visibilit�y��e,�h�and�ho�w�it�links�Mordell-����'�8W��eeil��<and�Shafarevic��!h-T�ate�groups,���ma��!y�pro�vide�new�insigh�t�in�to�implications�b�M�e-����'�8t��!w�een��fcases�of�the�Birc��!h�and�Swinnerton-Dy�er�conjecture.����'�8�Broader�2�Impact:����'�8�The��8PI��in��!tends�to�complete�the�undergraduate�textb�M�o�ok��8�p�0J
�3
cmsl10�Elemen�tary�Num�b�M�er�The-����'�8ory���and�Elliptic�Curv��!es�,��Awhic�h���he�is�writing�under�con��!tract�with�Springer-V��eerlag.����'�8He�Aalso�in��!tends�to�nish�the�graduate�textb�M�o�ok�A�Lectures�on�Mo�M�dular�F��eorms�and����'�8Hec��!k�e��*Op�M�erators�,��whic�h�he�is�co-authoring�with�Ken�Rib�M�et,��and�whic�h�is�lik�ely����'�8to��4b�M�e�published�b��!y�Springer-V��eerlag.���These�textb�o�oks�distinguish�themselv��!es�from����'�8similar�,Etitles�b��!y�incorp�M�orating�sp�ecic�kno��!wledge�and�in�tuition�gathered�b�y�the�PI����'�8from��fhis�past�n��!umerical�in�v�estigations.��P���'�8Con��!tin�ued��dev�elopmen�t�of�his�computational�programs�promises�to�ha�v�e�a�broader����'�8impact��on�n��!um�b�M�er��theory��e,�Wb�ecause�his�soft��!w�are�and�the�mo�M�dular�forms�database����'�8are��Nstandard�to�M�ols�for�obtaining�data�ab�out�mo�dular�forms�and�asso�ciated�ob���jects.����'�8Elliptic��Mcurv��!es�and�ab�M�elian�v��dDarieties�o�v�er�nite�elds�are�used�in�ev�eryda�y�public-����'�8k��!ey�7hcryptograph�y��e.���Though�no�sp�M�ecic�applications�to�cryptograph�y�are�giv�en�in����'�8this�e�prop�M�osal,�r�the�PI�e�hop�es�the�to�ols�and�data�that�he�dev��!elops�will�pro�vide�input����'�8to��fresearc��!hers�who�are�analyzing�and�designing�new�cryptosystems.�����������C�1���������*���;���>�g0	�m#�R
�3
cmss10�m#�R

cmss10�F
C�G�
cmbxti10�p�0J
�3
cmsl10��"V
�3
cmbx10�K�`y
�3
cmr10�
��������