Minimal Subgroups of ${\rm GL}_2(\mathbb{Z}_{S})$

Abstract

Let $E$ be an elliptic curve over a number field $L$ and for a finite set $S$ of primes, let $ρ_{E,S} : {\rm Gal}(\overline{L}/L) \to {\rm GL}_{2}(\mathbb{Z}_{S})$ be the $S$-adic Galois representation. If $L \cap \mathbb{Q}(ζ_{n}) = \mathbb{Q}$ for all positive integers $n$ whose prime factors are in $S$, then $\det ρ_{E,S} : {\rm Gal}(\overline{L}/L) \to \mathbb{Z}_{S}^{\times}$ is surjective. We say that a finite index subgroup $H \subseteq {\rm GL}_{2}(\mathbb{Z}_{S})$ 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 ${\rm GL}_{2}(\mathbb{Z}_{S})$ unless $S = \{ 2 \}$, while minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$ are plentiful. We give models for all the genus $0$ modular curves associated to minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$, and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at $2$ and with minimal $2$-adic image.