CoCalc Public Fileswww / nsf / project_summary.tex
Author: William A. Stein
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7\noindent{}The Birch and Swinnerton-Dyer conjecture and Mazur's
8notion of visibility of Shafarevich-Tate groups motivate the
9computational and theoretical goals of this proposal.  The
10computational goals are to develop new algorithms, tables, and
11software for studying modular forms and modular abelian varieties.
12The theoretical goals are to prove new theorems that relate
13Mordell-Weil and Shafarevich-Tate groups of elliptic curves and
14abelian varieties.
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18\noindent{\bf Intellectual Merit:}
19
20\noindent{}The PI is one of the more sought-after people by the
21worldwide community of number theorists, for computational
22confirmation of conjectures, for modular forms algorithms, for
23data, and for ways of formulating problems so as to make them more
24accessible to algorithms.   This project will provide new
25computational tools, including a major new package for computing
26with modular abelian varieties over number fields, and enhance the
27modular forms database, which is used by many mathematicians who
28study modular forms.
29
30\mypar{}The PI's investigations into Mazur's notion of visibility,
31and how it links Mordell-Weil and Shafarevich-Tate groups, may
32provide new insight into implications between cases of the Birch
33and Swinnerton-Dyer conjecture.
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39\noindent{}The PI intends to complete the undergraduate textbook
40{\sl Elementary Number Theory and Elliptic Curves}, which he is
41writing under contract with Springer-Verlag. He also intends to
42finish the graduate textbook {\sl Lectures on Modular Forms and
43Hecke Operators}, which he is co-authoring with Ken Ribet, and
45textbooks distinguish themselves from similar titles by
46incorporating specific knowledge and intuition gathered by the PI
47from his past numerical investigations.
48
49\mypar{}Continued development of his computational programs
50promises to have a broader impact on number theory, because his
51software and the modular forms database are standard tools for
52obtaining data about modular forms and associated objects.
53
54\mypar{}Elliptic curves and abelian varieties over finite fields
55are used in everyday public-key cryptography.  Though no specific
56applications to cryptography are given in this proposal, the PI
57hopes the tools and data that he develops will provide input to
58researchers who are analyzing and designing new cryptosystems.
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