Elliptic curves over a finite field and the trace formula

Abstract

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 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 Z/2Z and (Z/2Z) 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.

Weight enumerators of Reed-Muller codes from cubic curves and their duals

“…We consider set of smooth projective plane cubic curves C based on E(F q ) [3] for the elliptic curve E satisfying C ∼ = E. By inclusion-exclusion, we can do this by dividing these cubics into those for which E(F q ) [3] has a subgroup isomorphic to Z/3Z and those for which E(F q )[3] ∼ = Z/3Z × Z/3Z. Following the terminology from [18], let C(A 3,3 t) denote the set of isomorphism classes of elliptic curves E ∈ C with E(F q )[3] ∼ = Z/3Z × Z/3Z and #E(F q ) = q + 1 − t. The following result is a special case of a weighted version of [23,Theorem 4.9].…”

“…The author and Petrow give formulas for exactly these types of expressions in Theorem 3 of [18]. Stating the full result would require introducing too much additional notation, so we refer to [18] for details. The case where R is even and q = p is prime is addressed in [18,Examples 1 and 2].…”

“…Stating the full result would require introducing too much additional notation, so we refer to [18] for details. The case where R is even and q = p is prime is addressed in [18,Examples 1 and 2]. We see that E p (t 2R E Φ Z/3Z ) can be expressed in terms of the traces of the Hecke operator T p acting on spaces of cusp forms for Γ 1 (3) of weight at most 2R + 2, and that E p (t 2R E Φ Z/3Z×Z/3Z ) can be expressed in terms of traces of T p acting on spaces of cusp forms for Γ(3) of weight at most 2R + 2.…”

“…Formulas for these quantities when q is a prime power, or when R is odd, are more intricate. See [18,Theorem 3] for details.…”

On the trace formula for Hecke operators on congruence subgroups, II

“…After submitting this paper, I found out that a trace formula for 1 (N ), as well as for more general congruence subgroups, was obtained by Kaplan and Petrow [6], who derived it from Oésterle's trace formula assuming the index of the Hecke operator is coprime with the level.…”