Split-CM points and central values of Hecke L-series

Abstract

Split-CM points are points of the moduli space h_2/Sp_4(Z) corresponding to products $E\times E^{\prime }$ of elliptic curves with the same complex multiplication. We prove that the number of split-CM points in a given class of h_2/Sp_4(Z) is related to the coefficients of a weight 3/2 modular form studied by Eichler. The main application of this result is a formula for the central value $L({\psi }_N,1)$ of a certain Hecke L-series. The Hecke character ${\psi }_N$ is a twist of the canonical Hecke character $\psi$ for the elliptic Q-curve A studied by Gross, and formulas for $L(\psi ,1)$ as well as generalizations were proven by Villegas and Zagier. The formulas for $L(\psin, 1)$ are easily computable and numerical examples are given.