Hecke Characters and Formal Group Characters

Let E be an elliptic curve over Q with complex multiplication by the ring of integers of an imaginary quadratic eld. Let be the Hecke character associated to E by the theory of complex multiplication. Let be the complex conjugate character of. For a pair of integers k and j, deene the Hecke character ' = '(k; j) = k j. Let p be a prime where E has good, ordinary reduction. Let p be a xed prime of K dividing p. In this paper, we study properties of the p-adic Hecke character ' p. In section 1, we establish a relation identifying ' p with the character from a Lubin-Tate formal group. This extends classical results on the Hecke character associated to E and the p-adic cyclotomic character p = p p. Then, in section 2, we use this relation, together with results of Fontaine Fo2] and Harrison Ha], to give an explicit description of the exponential map of the p-adic Galois representation given by ' p. We then describe the image of the exponential map in section 3. Such an description of the exponential map is important in studying the arithmetic properties of the Hecke character '. For another approach to explicit exponential maps, see the papers of Kato Ka] and Han Han]. The current paper and Han's paper were written independently, and the contexts, results, as well as methods, of the two papers are diierent.