Elliptic curves over a finite field and the trace formula

TL;DR

This paper derives formulas for the moments of elliptic curve point counts over finite fields, linking them to traces of Hecke operators, and generalizes previous results through an Eichler-Selberg trace formula for various congruence subgroups.

Contribution

It introduces a new trace formula for congruence subgroups of SL_2(Z) and applies it to compute moments of elliptic curve point counts, extending prior work.

Findings

Formulas for power moments of elliptic curve point counts.

Connection between moments and traces of Hecke operators.

Generalization of previous results to broader subgroups.

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 $\Gamma_1(N)$ and $\Gamma(N)$. 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 $\Gamma_0(2),\Gamma_0(4)$). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vl\v{a}du\c{t}, Gekeler, Howe, and others.