Anticyclotomic p-adic L-function of central critical Rankin-Selberg L-value

TL;DR

This paper constructs a new p-adic L-function that interpolates central critical Rankin-Selberg L-values for automorphic forms over imaginary quadratic fields, extending classical results to a p-adic setting.

Contribution

It introduces a novel p-adic L-function for Rankin-Selberg L-values, generalizing Katz's interpolation for Eisenstein series to cusp forms.

Findings

Constructed a p-adic L-function interpolating L-values for automorphic forms.

Extended classical interpolation results to the Rankin-Selberg setting.

Provides a new tool for studying special values of L-functions in a p-adic context.

Abstract

Let M be an imaginary quadratic field, f a Hecke eigenform on GL2(Q) and \pi the unitary base-change to M of the automorphic representation associated to f. Take a unitary arithmetic Hecke character \chi of M inducing the inverse of the central character of f. The celebrated formula of Waldspurger relates the square of an integral L(f) of f and \chi over the idele class group of M to the central critical value L(1/2,{\pi}\otimes{\chi}). In this paper, we present a construction of a new p-adic L-function that interpolates L(f) over arithmetic characters for a cusp form f in the spirit of the landmark result of Katz where he did that for Eisenstein series.