Ross Paterson

Average Ranks of Elliptic Curves After p-Extension

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 progress on this question, answering it in the affirmative for certain choices of K. This progress follows from a statistical study of certain local invariants of elliptic curves, which loosely describe the failure of Galois descent for the associated p-Selmer groups. The main inputs come from modular representation theory, arithmetic duality, and good old-fashioned counting.