Quebec Mathematical Sciences Colloquium

January 27, 2023 from 15:30 to 16:30 (Montreal/EST time) On location

The distribution of Selmer groups of elliptic curves

Colloquium presented by Arul Shankar (University of Toronto)

The Goldfeld and Katz--Sarnak conjectures predict that 50% of elliptic curves have rank 0, that 50% have rank 1, and that the average rank of elliptic curves is 1/2 (the remaining 0% of elliptic curves not interfering in the average). Successive works of Brumer, Heath-Brown, and Young, have approached this problem by studying the central values of the L functions of elliptic curves. In this talk, we will take an algebraic approach, in which we study the ranks of elliptic curves via studying their Selmer groups.

Poonen and Stoll developed a beautiful model for the behaviours of $p$-Selmer groups of elliptic curves, and gave heuristics for all moments of the sizes of these groups.
In this talk, I will describe joint work with Manjul Bhargava and Ashvin Swaminathan, in which we prove that the second moment of the size of the 2-Selmer groups of elliptic curves is bounded above by 15 (which is the constant predicted by Poonen--Stoll).

There won’t be a Zoom broadcast for this conference


McGill University, Burnside Hall room 1205