Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves

Abstract

We determine asymptotic results for the average size of Selmer groups arising from certain isogenies related to Jacobians of hyperelliptic curves of genus $g\ge 2$. We do so by combining Bhargava's geometry-of-numbers methods with new parametrisations coming from Vinberg theory, arising from representations related to the Dynkin diagrams of type $B$ and $C$. We additionally prove some lower bounds on the average size of these isogeny Selmer groups by using a formula of Greenberg--Wiles.

35 pages