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arXiv · 2308.08663

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

Daniel Barrera Salazar·Ariel Pacetti·Gonzalo Tornaría
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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 $y^2 = x^5 + x$.

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Daniel Barrera Salazar, Ariel Pacetti, Gonzalo Tornaría. 2023-08-16. On the $2$-Selmer group of Jacobians of hyperelliptic curves. https://arxiv.org/abs/2308.08663

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