Two p-adic L-functions and rational points on elliptic curves with supersingular reduction

Let E be an elliptic curve over Q. We assume that E has good supersingular reduction at a prime p, and for simplicity, assume p is odd and ap = p+ 1− #E(Fp) is zero. Then, as the second author showed, the p-adic L-function Lp,α(E) of E corresponding to α = ±√−p (by Amice-Vélu and Vishik) can be written as Lp,α(E) = f log+p +g logp α by using two Iwasawa functions f and g ∈ Zp[[Gal(Q∞/Q)]] ([20] Theorem 5.1). Here logp is the ±-log function and Q∞/Q is the cyclotomic Zp-extension (precisely, see §1.3). In Iwasawa theory for elliptic curves, the case when p is a supersingular prime is usually regarded to be more complicated than the ordinary case, but the fact that we have two nice Iwasawa functions f and g gives us some advantage in several cases. The aim of this paper is to give such examples.