Lifting images of standard representations of symmetric groups

Abstract

We investigate closed subgroups G ⊆ Sp 2g (Z2) whose modulo-2 images coincide with the image S2g+1 ⊆ Sp 2g (F2) of S2g+1 or the image S2g+2 ⊆ Sp 2g (F2) of S2g+2 under the standard representation. We show that when g ≥ 2, the only closed subgroup G ⊆ Sp 2g (Z2) surjecting onto S2g+2 is its full inverse image in Sp 2g (Z2), while all subgroups G ⊆ Sp 2g (Z2) surjecting onto S2g+1 are open and contain the level-8 principal congruence subgroup of Sp 2g (Z2). 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.