Duke Mathematical Journal
- Duke Math. J.
- Volume 118, Number 3 (2003), 523-558.
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  and B. Perrin-Riou in  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.
Duke Math. J. Volume 118, Number 3 (2003), 523-558.
First available in Project Euclid: 23 April 2004
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Pollack, Robert. On the p -adic L -function of a modular form at a supersingular prime. Duke Math. J. 118 (2003), no. 3, 523--558. doi:10.1215/S0012-7094-03-11835-9. http://projecteuclid.org/euclid.dmj/1082744678.