On the 2-Selmer group of Jacobians of hyperelliptic curves

Abstract

Let C be a hyperelliptic curve y2=p(x) defined over a number field K with p(x) integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the 2-Selmer group of the Jacobian of C in terms of the class group of the K-algebra K[x]/(p(x)). Our main result is a formula relating these two quantities under some mild hypothesis. We provide some examples that prove that our lower and upper bounds are as sharp as possible. As a first application, we study the rank distribution of the 2-Selmer group in families of quadratic twists. Under some extra hypothesis we prove that among prime quadratic twists, a positive proportion has fixed 2-Selmer group. As a second application, we study the family of octic twists of the genus 2 curve y2=x5+x.