Minimal Subgroups of GL₂(ZS)

Let E be an elliptic curve over a number field K and for a finite set S of primes, let ρE,S : Gal(Q̄/Q) → GL₂(ZS) the S-adic Galois representation. We say that a finite index subgroup H ⊆ GL₂(ZS) is minimal if : H → ZS× is surjective, but : K → ZS× is not surjective for any proper closed subgroup K of H. We show that there are no minimal subgroups of GL₂(ZS) unless S = \ 2 \, while minimal subgroups of GL₂(Z₂) are plentiful. We give models for all genus $0$ minimal subgroups of GL₂(Z₂), and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at $2$ and with minimal $2$-adic image.