Supersingular primes and p-adic L-functions

We discuss the problem of finding a p-adic L-function attached 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 essentially understood, yielding many deep and interesting results, basic questions remain unanswered in the case of supersingular reduction. We will discuss a conjecture, related to another in Rubin, 1987, and some ideas related to the problem in general. 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.