On the p-ADIC L-function of a modular form at a supersingular prime

In this paper we study the two p-adic L-functions attached to a modular form f = Sigmaa(n)q(n) at a supersingular prime p. When a(p) = 0, we are able to decompose both the sum and the difference of the two unbounded distributions attached to f into a bounded measure and a distribution that accounts for all of the growth. Moreover, this distribution depends only upon the weight of f (and the fact that a p vanishes). From this description we explain how the p-adic L-function is controlled by two Iwasawa functions and by two power series with growth which have a fixed infinite set of zeros (Theorem 5.1). Asymptotic formulas-for-the p-part of the analytic size of the Tate-Shafarevich group of an elliptic curve in the cyclotomic direction are computed, using this result. These formulas compare favorably with results established by M. Kurihara in [11] and B. Perrin-Riou in [23] on the algebraic side. Moreover, we interpret Kurihara's conjectures on the Galois structure of the Tate-Shafarevich group in terms of these two Iwasawa functions.