The Gross-Zagier Formula and Kolyvagin's Conjecture for Elliptic Curves of Higher Rank

Proposal Number: #7074456

Program Solicitation

William Stein


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.

Project Summary PDF, TEX
Project Description PDF, TEX
References Cited PDF, TEX
Stein Biographical Sketch PDF, TEX
Budget Justification PDF, TEX
Complete Tarball 2010-ant.tar.bz2