Number Theory Seminar

In 1979 Goldfeld conjectured that, given an elliptic curve E over the rationals, 50% of the quadratic twists of E have analytic rank 0, and 50% have analytic rank 1. In this talk, we show that this conjecture is implied by the Birch and Swinnerton-Dyer conjecture. Our approach to this problem is to determine the distribution of 2^k-Selmer groups in the quadratic twist family of E. A potential obstacle to our work is that the distribution of 2-Selmer groups in this twist family depends delicately on E. We talk about some of the unusual 2-Selmer distributions that appear, and explain how such behavior impacts our work on 4-, 8-, 16- ... Selmer ranks.