Minimal Subgroups of ${\mathrm{GL}}_2({\mathbb{Z}}_S)$

Abstract

Let $E$ be an elliptic curve over a number field $L$ and for a finite set $S$ of primes, let $\ddot{I}{}_{E,S}:\mathrm{Gal}(\overline{L}/L)\to {\mathrm{GL}}_2({\mathbb{Z}}_S)$ be the $S$-adic Galois representation. If $L\cap \mathbb{Q}(\hat{I}{\P }_n)=\mathbb{Q}$ for all positive integers $n$ whose prime factors are in $S$, then $\det \ddot{I}{}_{E,S}:\mathrm{Gal}(\overline{L}/L)\to {\mathbb{Z}}_S^{\times }$ is surjective. We say that a finite index subgroup $H\subseteq {\mathrm{GL}}_2({\mathbb{Z}}_S)$ is minimal if $\det :H\to {\mathbb{Z}}_S^{\times }$ is surjective, but $\det :K\to {\mathbb{Z}}_S^{\times }$ is not surjective for any proper closed subgroup $K$ of $H$. We show that there are no minimal subgroups of ${\mathrm{GL}}_2({\mathbb{Z}}_S)$ unless $S=\{2\}$, while minimal subgroups of ${\mathrm{GL}}_2({\mathbb{Z}}_2)$ are plentiful. We give models for all the genus $0$ modular curves associated to minimal subgroups of ${\mathrm{GL}}_2({\mathbb{Z}}_2)$, and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at $2$ and with minimal $2$-adic image.

15 pages