# Boundedness results for 2-adic Galois images associated to hyperelliptic Jacobians

## TL;DR

This paper establishes boundedness results for 2-adic Galois images of hyperelliptic Jacobians over number fields, showing under certain conditions the images are open and contain specific congruence subgroups, with implications for uniform boundedness.

## Contribution

It provides new criteria ensuring the openness of 2-adic Galois images for hyperelliptic Jacobians and extends uniform boundedness results to a broader class of curves.

## Key findings

- Galois images are open under specific prime divisibility conditions.
- Almost all parameters satisfy the key hypothesis for boundedness.
- Quantitative bounds are established for Galois image sizes.

## Abstract

Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of discriminant $\Delta$ and some element $\lambda \in \mathcal{O}_{K}$. Our first main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(\lambda))$ but not $(2\Delta)$, then the image of the natural $2$-adic Galois representation is open in $\mathrm{GSp}(T_{2}(J))$ and contains a certain congruence subgroup of $\mathrm{Sp}(T_{2}(J))$ depending on the maximal power of $\mathfrak{p}$ dividing $(f(\lambda))$. We also present and prove a variant of this result that applies when $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)(x - \lambda')$ for distinct elements $\lambda, \lambda' \in K$. We then show that the hypothesis in the former statement holds for almost all $\lambda \in \mathcal{O}_{K}$ and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.