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arXiv · 1510.03980

Elliptic curves over a finite field and the trace formula

Nathan Kaplan·Ian Petrow
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Abstract

We prove formulas for power moments for point counts of elliptic curves over a finite field $k$ such that the groups of $k$-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$. As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$ which include as special cases the groups $Γ_1(N)$ and $Γ(N)$. Our formulas generalize results of Birch and Ihara (the case of the trivial subgroup, and the full modular group), and previous work of the authors (the subgroups $\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{Z}/2\mathbb{Z})^2$ and congruence subgroups $Γ_0(2),Γ_0(4)$). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vlǎduţ, Gekeler, Howe, and others.

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Nathan Kaplan, Ian Petrow. 2017-08-08. Elliptic curves over a finite field and the trace formula. https://doi.org/10.1112/plms.12069

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