Fourier expansion of Arakawa lifting II: Relation with central L-values

TL;DR

This paper investigates the Fourier coefficients of Arakawa lifts and their explicit relation to central values of automorphic L-functions, providing formulas and conditions for positivity.

Contribution

It explicitly relates Fourier coefficients of Arakawa lifts to central L-values and determines the proportionality constants, extending previous work on automorphic forms.

Findings

Explicit formulas for Fourier coefficients in terms of L-values

Relation between Fourier coefficient norms and central L-values

Discussion on positivity of central L-values

Abstract

This is a continuation of our previous paper published in Israel J. Math 187 (2012), 317-369. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic L-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central $L$-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central L-value. We can relate the square norm with the central value of some $L$-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the L-functions in our concern.