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= (antegral) anq^n at a supersingular prime p. When ap=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 AP 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 TateShafarevich 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 Kuriharaʹs conjectures on the Galois structure of the Tate-Shafarevich group in terms of these two Iwasawa functions.