A p-adic analytic family of the D-th Shintani lifting for a Coleman family and congruences between the central L-values

Abstract

We will construct a p-adic analytic family of D-th Shintani lifting generalized by Kojima and Tokuno for a Coleman family. Consequently, we will have a p-adic L-function which interpolates the central L-values attached to a Coleman family and obtain a congruence between the central L-values. Focusing on the case of p-ordinary, we will obtain two applications. One of them states that a congruence between Hecke eigenforms of different weights sufficiently close, p-adically, derives a congruence between their central L-values. The other one is about the Goldfeld conjecture in analytic number theory. We will show that there exists a primitive form satisfying the conjecture for each even weight sufficiently close to 2, 3-adically, thanks to a result of Vatsal.