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

2010/05/07 by Kimberly Hopkins, Hopkins, Kimberly
Mathematics · #11 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11

paper · pdf · doi:10.48550/arxiv.1005.1240

arxiv created 2010/05/07 · openalex publication_date 2010/05/07 · arxiv updated 2010/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Split-CM points are points of the moduli space h₂/Sp₄(Z) corresponding to products E × E' of elliptic curves with the same complex multiplication. We prove that the number of split-CM points in a given class of h₂/Sp₄(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(ψ_N, 1) of a certain Hecke L-series. The Hecke character ψ_N is a twist of the canonical Hecke character ψ for the elliptic Q-curve A studied by Gross, and formulas for L(ψ, 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.

Citations

• On Modular Forms of Half Integral Weight

Related

• A formula for the central value of certain Hecke L-functions
• Distribution of root numbers of Hecke characters attached to some elliptic curves
• A generalized Rubin formula for Hecke characters
• Non-vanishing of the Central Derivative of Canonical Hecke L-functions
• Quasi-modular forms attached to elliptic curves: Hecke operators