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Author: William A. Stein
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\noindent{}The Birch and Swinnerton-Dyer conjecture and Mazur's
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notion of visibility of Shafarevich-Tate groups motivate the
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computational and theoretical goals of this proposal. The
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computational goals are to develop new algorithms, tables, and
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software for studying modular forms and modular abelian varieties.
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The theoretical goals are to prove new theorems that relate
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Mordell-Weil and Shafarevich-Tate groups of elliptic curves and
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abelian varieties.
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\noindent{\bf Intellectual Merit:}
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\noindent{}The PI is one of the more sought-after people by the
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worldwide community of number theorists, for computational
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confirmation of conjectures, for modular forms algorithms, for
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data, and for ways of formulating problems so as to make them more
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accessible to algorithms. This project will provide new
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computational tools, including a major new package for computing
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with modular abelian varieties over number fields, and enhance the
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modular forms database, which is used by many mathematicians who
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study modular forms.
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\mypar{}The PI's investigations into Mazur's notion of visibility,
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and how it links Mordell-Weil and Shafarevich-Tate groups, may
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provide new insight into implications between cases of the Birch
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and Swinnerton-Dyer conjecture.
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\noindent{\bf Broader Impact:}
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\noindent{}The PI intends to complete the undergraduate textbook
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{\sl Elementary Number Theory and Elliptic Curves}, which he is
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writing under contract with Springer-Verlag. He also intends to
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finish the graduate textbook {\sl Lectures on Modular Forms and
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Hecke Operators}, which he is co-authoring with Ken Ribet, and
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which is likely to be published by Springer-Verlag. These
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textbooks distinguish themselves from similar titles by
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incorporating specific knowledge and intuition gathered by the PI
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from his past numerical investigations.
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\mypar{}Continued development of his computational programs
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promises to have a broader impact on number theory, because his
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software and the modular forms database are standard tools for
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obtaining data about modular forms and associated objects.
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\mypar{}Elliptic curves and abelian varieties over finite fields
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are used in everyday public-key cryptography. Though no specific
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applications to cryptography are given in this proposal, the PI
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hopes the tools and data that he develops will provide input to
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researchers who are analyzing and designing new cryptosystems.
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