Algebra Number Theory 16 (4), 777-837, (2022) DOI: 10.2140/ant.2022.16.777
KEYWORDS: Elliptic curve, Complex Multiplication, p-adic, Galois representation, Cartan, 11F80, 11G05, 11G15, 14H52
We give an explicit classification of the possible -adic Galois representations that are attached to elliptic curves with CM defined over . More precisely, let be an imaginary quadratic field, and let be an order in of conductor . Let be an elliptic curve with CM by , such that is defined by a model over . Let be a prime, let be the absolute Galois group of , and let be the Galois representation associated to the Galois action on the Tate module . The goal is then to describe, explicitly, the groups of that can occur as images of , up to conjugation, for an arbitrary order .