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.