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arXiv · 1005.1240

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

Kimberly Hopkins
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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 $L(ψ_N, 1)$ 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.

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Kimberly Hopkins. 2010-05-07. Split-CM points and central values of Hecke L-series. https://arxiv.org/abs/1005.1240

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