On elliptic units and p-adic Galois representations attached to elliptic curves

AbstractLet K be a quadratic imaginary number field with discriminant DK≠-3,-4 and class number one. Fix a prime p⩾7 which is not ramified in K and write hp for the class number of the ray class field of K of conductor p. Given an elliptic curve A/K with complex multiplication by K, let ρA¯:Gal(K¯/K(μp∞))→SL(2,Zp) be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if p∤hp then the image of a certain deformation ρA:Gal(K¯/K(μp∞))→SL(2,Zp[[X]]) of ρA¯ is “as big as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Zp). The proof rests on the theory of Siegel functions and elliptic units as developed by Kubert, Lang and Robert