| In this paper, we study the p-adic L-functions attached to a modular form f = sumanq n at a supersingular prime and mainly the case when ap = 0. It is known in many cases that these L-functions have infinitely many zeroes (in the "extended disc"). Therefore, the zeroes are not controlled by a single polynomial in the Iwasawa algebra as in the ordinary case. The main result of this paper (Theorem 6.1) describes how the zeroes of these L-functions are controlled by two polynomials and by two "gamma-like" functions each with a fixed infinite set of trivial zeroes. Also, 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 Kurihara in [9] on the algebraic side. |
