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### arXiv:1903.06148 [math.NT]AbstractReferencesReviewsResources

#### Lifting images of standard representations of symmetric groups

Published 2019-03-14Version 1

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 the only closed subgroup $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ surjecting onto $\mathfrak{S}_{2g + 2}$ is its full inverse image in $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$, and we classify the modulo-$4$ images of all closed subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ surjecting onto $\mathfrak{S}_{2g + 1}$. We also prove a corollary concerning even-degree polynomials with full Galois group which is very elementary to state.

Derivative of the standard $p$-adic $L$-function associated with a Siegel form