On elliptic units andp-adic Galois representations attached to elliptic curves

Fix a prime p ≥ 7 and a number field K, and write K&d15; for the extension of K generated by the roots of unity in K¯ of p-power order. Given an elliptic curve E over K(j) with transcendental invariant j, D. Rohrlich constructs a Galois representation rE : Gal Kj /K&d15;j → SL(2, Zp X ) such that the representation rE |X = 0 is equivalent to the natural representation of Gal Kj /K&d15;j on Tp(E), the Tate module of E. Let A be an elliptic curve over K with j(A) ≠ 0, 1728 and suppose that A coincides with the fiber of E at j = j(A). Then rE can be restricted to the decomposition group corresponding to a place extending j = j(A) of K&d15;j , to obtain a representation rA : Gal( K/K&d15; ) → SL(2, Zp X ). Let rA : Gal( K/K&d15; ) → SL(2, Fp ) be the representation induced by the action of Galois on the points of order p on A. In light of the results of Serre on the properties of the image of rA , in particular the fact that rA is surjective for all sufficiently large p, we would like to study the image of rA . Rohrlich proved that, for an elliptic curve A over Q , if rA is surjective and nup(j( A)) = -1 then rA is surjective.; The first part of the dissertation is devoted to proving the following generalization of the result to arbitrary number fields. Let A/K be an elliptic curve as above. Fix ℘, a prime of K lying above p. We write nu℘ for the standard ℘-adic valuation on K, and e = e(℘ | p) for the ramification index.; Theorem A. If rA is surjective, e is not divisible by p - 1, nu℘(j(A)) = - t with t ∈ N , gcd(p,t) = 1, and t < epp-1=e+ep-1 , then rA is surjective.; The main result of the dissertation comes from the study of the image of rA in the case of elliptic curves with complex multiplication, using the work of G. Robert, D. Kubert and S. Lang on elliptic units.; Let K be a quadratic imaginary number field with discriminant DK ≠ -3, -4 and class number h K = 1. Fix a prime p ≥ 7 which is not ramified in K. Given an elliptic curve A/K with complex multiplication by K (and precisely by the ring of integers OK ), let rA : Gal( K/K&d15; ) → SL(2, Zp ) be the representation determined up to equivalence by the action of Gal( K/K&d15; ) on Tp(A). By the theory of complex multiplication, the image of this map is a Cartan subgroup C' of SL(2, Zp ). We write K(p) for the ray class field of K of conductor pOK , and we let hp be the class number of K(p).; Theorem B. If p ∤ hp then rAGal&parl0; K/K&d5; &parr0; is "as big as possible": that is, it is the full inverse image of C' under the natural map piX : SL(2, Zp X ) → SL(2, Zp ).