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 $g\geq 2$. 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.