Central critical values of modular Hecke L-functions

Abstract

We give an explicit formula for the central critical value $L(1/2,\mathrm{\pi ̂}\otimes \chi )$ of the base-change lift $\mathrm{\pi ̂}$ to an imaginary quadratic field $K$ of an automorphic representation $\pi$ as the square of a finite sum of the values of a nearly holomorphic cusp form in $\pi$ at elliptic curves with complex multiplication by $K$. As long as the transcendental factor of the value is a CM period, $\chi$ is basically any unitary arithmetic Hecke character of $K$ inducing the inverse of the central character of $\pi$.