NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	UNIVERSITY OF CALIFORNIA, SAN DIEGO
Initial Amendment Date:	November 29, 2005
Latest Amendment Date:	April 14, 2006
Award Number:	0555776
Award Instrument:	Continuing Grant
Program Manager:	Tie Luo tluo@nsf.gov  (703)292-8448 DMS  Division Of Mathematical Sciences MPS  Direct For Mathematical & Physical Scien
Start Date:	July 1, 2005
End Date:	May 31, 2007 (Estimated)
Total Intended Award Amount:	$123,418.00
Total Awarded Amount to Date:	$123,418.00
Funds Obligated to Date:	FY 2004 = $19,742.00 FY 2005 = $28,074.00 FY 2006 = $0.00
History of Investigator:	William Stein (Principal Investigator) wstein@math.washington.edu
Recipient Sponsored Research Office:	University of California-San Diego 9500 GILMAN DR LA JOLLA CA  US  92093-0021 (858)534-4896
Sponsor Congressional District:	50
Primary Place of Performance:	University of California-San Diego 9500 GILMAN DR LA JOLLA CA  US  92093-0021
Primary Place of Performance Congressional District:	50
Unique Entity Identifier (UEI):	UYTTZT6G9DT1
Parent UEI:
NSF Program(s):	ALGEBRA,NUMBER THEORY,AND COM, COMPUTATIONAL MATHEMATICS
Primary Program Source:
Program Reference Code(s):	0000, 9263, OTHR
Program Element Code(s):	126400, 127100
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

ABSTRACT for the award DMS-0400386 of Stein

The Birch and Swinnerton-Dyer conjecture and Mazur's notion of
visibility of Shafarevich-Tate groups motivate the computational and
theoretical goals of this research project. The computational goals
are to develop new algorithms, tables, and software for studying
modular forms and modular abelian varieties. The PI hopes to create
new computational tools, including a major new package for computing
with modular abelian varieties over number fields, and enhance the
modular forms database, which is used by many mathematicians who study
modular forms. The theoretical goals are to prove new theorems that
relate Mordell-Weil and Shafarevich-Tate groups of elliptic curves and
abelian varieties. These investigations into Mazur's notion of
visibility, and how it links Mordell-Weil and Shafarevich-Tate groups,
may provide new insight into relationships between different cases of
the Birch and Swinnerton-Dyer conjecture.

Elliptic curves and modular forms play a central role in modern number
theory and arithmetic geometry. For example, Andrew Wiles proved
Fermat's Last Theorem by showing that the elliptic curve attached by
Gerhard Frey to a counterexample to Fermat's claim would be attached
to a modular form, and that this modular form cannot exist. Our
understanding of the world of elliptic curves and modular forms is
extensive, but many questions remain unresolved. The first goal of
this project is to provide theoretical and computational tools to make
modular forms and objects attached to them very explicit, so that
mathematicians can compute with them, test their conjectures on
them, and gain a better feeling for them. The second goal is to use
these tools and other ideas to gain a deeper understanding of the
conjecture of Bryan Birch and Peter Swinnerton-Dyer about the arithmetic of elliptic curves.

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