Lifting images of standard representations of symmetric groups

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 $g\ge 2$, 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.

arXiv comment23 pages, 6 sections, 14 works cited. This final update includes a number of minor corrections and is the exact version that has been published in Manuscripta Mathematica