2-Selmer groups of even hyperelliptic curves over function fields

Abstract

In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg's representation of the group G = PSO(2n+2) and a Hitchin fibration. Consistent with the result over Q of Arul Shankar and Xiaoheng Wang [Rational points on hyperelliptic curves having a marked non-Weierstrass point, in Compos. Math., 154(1):188-222, 2018], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperelliptic curves, we obtain precisely number 6.