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Average Ranks of Elliptic Curves After p-Extension

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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. Time permitting, we will also discuss upper and lower bounds for the average dimension of 2-Selmer groups over multi-quadratic extensions.

This talk is part of the Number Theory Seminar series.

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