Average Ranks of Elliptic Curves after p-Extension

Ross Paterson (U of Glasgow)

Sep 22. 2021, 09:15 — 10:15

As E varies among elliptic curves defined over the rational numbers, a theorem of Bhargava and Shankar shows that the average rank of the Mordell--Weil group E(Q) is bounded.  If we now fix a number field K, is the same true of E(K)?  I will report on recent progress on this question, answering this question in the affirmative for certain choices of K.  This progress follows from a study of certain local invariants of elliptic curves, which loosely describe the failure of Galois descent for the associated p-Selmer groups.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Arithmetic Statistics and Local-Global Principles (Workshop)
Tim Browning (ISTA, Klosterneuburg)
Daniel Loughran (U Bath)
Rachel Newton (KCL, London)