On the $2$-Selmer group of Jacobians of hyperelliptic curves

TL;DR

This paper establishes bounds for the 2-Selmer group of Jacobians of hyperelliptic curves over number fields, relating them to class groups, and explores their implications for rank distributions in families of quadratic and octic twists.

Contribution

It provides a formula linking the 2-Selmer group to class groups of associated algebras and applies this to analyze rank distributions in specific twist families.

Findings

Bounds for the 2-Selmer group are as sharp as possible.

A positive proportion of prime quadratic twists have a fixed 2-Selmer group.

Analysis of octic twists of a specific genus 2 curve.

Abstract

Let $\mathcal{C}$ be a hyperelliptic curve $y^2 = p(x)$ 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 $\mathcal{C}$ 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 $y^2 = x^5 + x$.