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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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