Bernoulli–Hurwitz numbers, Wieferich primes and Galois representations

AbstractLet K be a quadratic imaginary number field with discriminant DK≠−3,−4 and class number one. Fix a prime p⩾7 which is unramified in K. Given an elliptic curve A/Q 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, for all but finitely many inert primes p, the image of a certain deformation ρA:Gal(K¯/K(μp∞))→SL(2,Zp〚X〛) of ρA¯ is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Zp). If p splits in K, then the same result holds as long as a certain Bernoulli–Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert–Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields