Average size of 2-Selmer groups of Jacobians of hyperelliptic curves over function fields

TL;DR

This paper computes the average size of 2-Selmer groups for two families of hyperelliptic curves over function fields using a geometric method, extending previous techniques from elliptic curves.

Contribution

It introduces a geometric approach to determine the average 2-Selmer group sizes for hyperelliptic curves, generalizing prior methods from elliptic curves.

Findings

Average size of 2-Selmer groups computed for hyperelliptic curves

Method extends previous elliptic curve techniques

Results applicable over function fields

Abstract

In this paper, we are going to compute the average size of 2-Selmer groups of two families of hyperelliptic curves with marked points over function fields. The result will be obtained by a geometric method which could be considered as a generalization of the one that was used previously by Q.P. Ho, V.B. Le Hung, and B.C. Ngo to obtain the average size of 2-Selmer groups of elliptic curves.