Number Theory Seminar: The Average Size of 2-Selmer Groups of Elliptic Curves over Function Fields

SEMINARS, NUMBER THEORY

Speaker:

Niven Achenjang - MIT

Given an elliptic curve E over a global field K the abelian group E(K) is finitely generated, and so much effort has been put into trying to understand the behavior of E(K), as E varies. Of note, it is a folklore conjecture that, when all elliptic curves E/K are ordered by a suitably defined height, the average value of their ranks is exactly 1/2. One fruitful avenue for understanding the distribution of E(K) has been to first understand the distribution of the sizes of Selmer groups of elliptic curves. In this direction, various authors (including Bhargava-Shankar, Poonen-Rains, and Bhargava-Kane-Lenstra-Poonen-Rains) have made conjectures which predict, for example, that the average size of the n-Selmer group of E/K is equal to the sum of the divisors of n. In this talk, I will report on some recent work verifying this average size prediction, "up to small error term," whenever n=2 and K is any global *function* field. Results along these lines were previously known whenever K was a number field or function field of characteristic  >5, so the novelty of my work is that it applies even in "bad" characteristic.

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html