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.