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

摘要

Let K be a quadratic imaginary number field with discriminant D-K not equal -3. -4 and class number one. Fix a prime p >= 7 which is not ramified in K and write h p 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 P-A: Gal((K) over bar /K (mu(p)infinity)) -> SL(2, Z(p)) be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if p inverted iota hp then the image of a certain deformation P-A: Gal((K) over bar /K(mu(p)infinity)) -> SL(2, Z(p)[[X]]) of P-A 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. (c) 2005 Elsevier Inc. All rights reserved.