# Lifting images of standard representations of symmetric groups

## TL;DR

This paper classifies certain subgroups of symplectic groups over 2-adic integers related to symmetric groups and applies these results to Galois representations of hyperelliptic curves and polynomials.

## Contribution

It provides a complete description of subgroups lifting symmetric groups in symplectic groups over 2-adic integers, strengthening existing results on Galois representations.

## Key findings

- Unique subgroup lifting $rak{S}_{2g+2}$ is the full inverse image in $ ext{Sp}_{2g}(Z_2)$.
- Subgroups lifting $rak{S}_{2g+1}$ are open and contain a principal congruence subgroup.
- Application to hyperelliptic curves improves previous Galois representation results.

## Abstract

We investigate closed subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose modulo-$2$ images coincide with the image $\mathfrak{S}_{2g + 1} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 1}$ or the image $\mathfrak{S}_{2g + 2} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 2}$ under the standard representation. We show that when $g \geq 2$, the only closed subgroup $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ surjecting onto $\mathfrak{S}_{2g + 2}$ is its full inverse image in $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$, while all subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ surjecting onto $\mathfrak{S}_{2g + 1}$ are open and contain the level-$8$ principal congruence subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$. As an immediate application, we are able to strengthen a result of Zarhin on $2$-adic Galois representations associated to hyperelliptic curves. We also prove an elementary corollary concerning even-degree polynomials with full Galois group.