2-Selmer groups of even hyperelliptic curves over function fields

In this paper, we are going to compute the average size of 2-Selmer groups families even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on Vinberg’s representation group G = PSO ( 2 n + stretchy="false">) G=\text {PSO}(2n+2) and Hitchin fibration. Consistent with alttext="double-struck upper Q"> Q encoding="application/x-tex">\mathbb {Q} Arul Shankar Xiaoheng Wang [Compos. Math. 154 (2018), pp. 188–222], provide an bound lower average. However, if restrict family transversal curves, obtain precisely number 6.