Elliptic curves over a finite field and the trace formula

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.