The geometric distribution of Selmer groups over function fields

ALGEBRAIC GEOMETRY & NUMBER THEORY SEMINAR

Many interesting aspects of the arithmetic of elliptic curves over global fields are governed by Selmer groups, which are cohomological approximations to the group of rational points. The statistical behavior of Selmer groups has been the focus of much recent study, and there is a wide gap between what we can prove and what we believe is true. On the one hand, work of Bhargava and Shankar computes the average size of 2,3,4, and 5-Selmer groups. On the other hand, Bhargava-Kane-Lenstra-Poonen-Rains conjecture a precise distribution for n-Selmer groups, for any n. I will talk about a limiting situation, in the function field context, where the BKLPR distribution can actually be proved to model the distribution of Selmer groups. This is joint work with Aaron Landesman and Eric Rains.