Kaplan, N; Petrow, I; (2017) Elliptic curves over a finite field and the trace formula. Proceedings of the London Mathematical Society , 115 (6) pp. 1317-1372. 10.1112/plms.12069.

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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 SL 2(Z). As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of SL 2(Z) which include as special cases the groups Γ1(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 Z/2Z and (Z/2Z)2 and congruence subgroups Γ0(2),Γ0(4)). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vladuţ, Gekeler, Howe and others.

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