Two variable p-adic L functions attached to eigenfamilies of positive slope

Abstract

Consider a classical cusp eigenform f=∑n=1∞an(f)qn of weight k≥2 for Γ0(N) with a Dirichlet character ψ mod N, and let Lf(s,χ)=∑n=1∞χ(n)an(f)n-s denote the L-function of f twisted with an arbitrary Dirichlet character χ. For a prime number p≥5, consider a family of cusp eigenforms f(k′) of weight k′, k′↦{f(k′)= ∑n=1∞an(f(k′))qn} containing f=f(k), such that the Fourier coefficients an(f(k′)) are given by certain p-adic analytic functions k′↦an(f(k′)). The purpose of this paper is to construct a two variable p-adic L function attached to Coleman’s family {f(k′)} of cusp eigenforms of a fixed positive slope σ=vp(αp)>0 where αp=αp(k′) is an eigenvalue (which depends on k′) of the Atkin operator U=Up. Our p-adic L-function interpolates the special values Lf(k′)(s,χ) at points (s,k′) with s=1,2,...,k′-1. We give a construction using the Rankin-Selberg method and the theory of p-adic integration on a profinite group Y with values in an affinoid K-algebra A, where K is a fixed finite extension of Qp. Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. In their turn, such measures come from Eisenstein distributions with values in certain Banach A-modules M†=M†(N;A) of families of overconvergent forms over A.