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 $f=\sum a\sb nq\sp n$ at a supersingular prime $p$. When $a\sb p=0$, 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 $a\sb p$ 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.