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\Large\bf
The Gross-Zagier Formula and Kolyvagin's Conjecture for
Elliptic Curves of Higher Rank
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In the 1960s, based on extensive numerical evidence, Birch and
Swinnerton-Dyer conjectured that the algebraic and analytic ranks of
any elliptic curve are equal, where the analytic rank is the order of
vanishing of the associated Hasse-Weil $L$-function at 1. Their
conjecture is proved for elliptic curves over the rational numbers when
the analytic rank is at most 1, but little progress has been made when
the rank is at least 2. The PI intends to explore three approaches to
better understanding the conjecture when the rank is at least 2. The
first approach involves a conjecture of Kolyvagin about Heegner
points; the PI intends to verify the conjecture in specific cases for
elliptic curves of rank at least 2 by explicitly computing cohomology
classes, and prove results about how the cohomology classes are
distributed. The second strategy involves the Gross-Zagier formula,
where the PI intends to create new conjectural generalizations of the
formula to higher rank, motivated by results and conjectures of
Kolyvagin and others. The third strategy introduces elliptic curves
over totally real fields; here the PI intends to compute tables,
especially about elliptic curves of rank at least 2 and bounded
conductor over totally real fields, generalize the other steps to
totally real fields, and scrutinize cases in which the parameterizing
Shimura curve has small genus.
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\noindent{}{\bf Intellectual Merit:} The proposed research could
shed light on the Birch and Swinnerton-Dyer Conjecture, which is one of the central
problems in number theory, e.g., it was chosen by the Clay
Mathematics Institute as the Millennium Prize Problem in algebraic
number theory. Explicit work in the 1960s and 1970s by Birch,
Swinnerton-Dyer, Buhler, Stephens, Atkin, and others provided
critical insight on which some of the great triumphs of
Gross-Zagier, Kolyvagin, Wiles, and others in the 1980s and 1990s
were based.
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\noindent{}{\bf Broader Impact:} The PI is co-authoring a popular
expository book with Barry Mazur on the Riemann Hypothesis,
co-authoring an advanced graduate level book with Kenneth Ribet on
modular forms and Hecke operators, and intends to prepare a new
edition of his AMS book on computing with modular forms. He has
many tables of data that are freely available online, and whose
creation has been supported by NSF FRG grant DMS-0757627, and the
proposed research would expand these tables further. He will also
continue to organize the development of the NSF-funded open source
Sage mathematical software project that he started. The PI
organizes dozens of workshops that involve many graduate students,
that have a potentially broad impact on the mathematics community.
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