Supersingular primes and p-adic L-functions

Abstract

We discuss the problem of finding a p-adic L-function at-tached to an elliptic curve with complex multiplication over an imaginary quadratic field K, for the case of a prime where the curve has supersingular reduction. While the case of primes of ordinary reduction has been extensively studied and is essen-tially understood, yielding many deep and interesting results, basic questions remain unanswered in the case of supersingu-lar reduction. We will discuss a conjecture, related to another in Rubin, 1987, and some ideas related to the problem in gen-eral. The basic tools originate with the work of J. Coates and A. Wiles in 1977 and 1978, and are developed in the work of K. Rubin. 1. Set-up. The analytic theory of L-functions and arithmetic properties of their special values goes back to the 19th-century work of Kummer on the arithmetic of