# Mathematical Sciences Research Institute

Home » DDC - Diophantine Problems: $2^k$-Selmer groups, the Cassels-Tate pairing, and Goldfeld's conjecture

# Seminar

DDC - Diophantine Problems: $2^k$-Selmer groups, the Cassels-Tate pairing, and Goldfeld's conjecture September 28, 2020 (09:00 AM PDT - 10:00 AM PDT)
Parent Program: Decidability, definability and computability in number theory: Part 1 - Virtual Semester MSRI: Online/Virtual
Speaker(s) Alexander Smith (Massachusetts Institute of Technology)
Description

This seminar will focus on Diophantine problems in a broad sense, with a view towards (but not limited to) interactions between Number Theory and Logic. Particular attention will be given to topics with the potential of further developments in the context of this MSRI scientific program. This will provide an opportunity for researchers to update on new results, techniques and some of the main problems of the field.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Video

#### $2^K$-Selmer Groups, The Cassels-Tate Pairing, And Goldfeld's Conjecture

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Abstract: Take $E$ to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that $100\%$ of the quadratic twists of $E$ have rank at most one. To do this, we will find the distribution of $2^k$-Selmer ranks in this family for every $k > 1$.

 Notes 263 KB application/pdf

#### $2^K$-Selmer Groups, The Cassels-Tate Pairing, And Goldfeld's Conjecture

 H.264 Video 25169_28667_8522__2_k_-Selmer_groups_The_Cassels-Tate_pairing__and_Goldfeld's_Conjecture.mp4