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

TL;DR

This paper links split-CM points on moduli space to modular form coefficients and derives a formula for the central value of a specific Hecke L-series, with computational examples.

Contribution

It establishes a relationship between split-CM points and modular form coefficients, providing a new formula for Hecke L-series central values.

Findings

Number of split-CM points relates to weight 3/2 modular form coefficients.

Derived a formula for the central value L(ψ_N, 1).

Provided numerical examples demonstrating the formula's computability.

Abstract

Split-CM points are points of the moduli space h_2/Sp_4(Z) corresponding to products $E \times E'$ 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.