Generating functions for power moments of elliptic curves over \mathbbF_p

2018/07/02 by Katherine Gallagher, Gallagher, Katherine, Lucia M. Li +7
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1807.00749

openalex publication_date 2018/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Seminal works by Birch and Ihara gave formulas for the mth power moments of the traces of Frobenius endomorphisms of elliptic curves over \mathbbF_p for primes p ≥ 5. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a fixed finite abelian group A. We revisit these formulas and determine a simple expression for the zeta function Z_p(A; t), the generating function for these mth power moments. In particular, we find that Z_p(A;t) = \frac\widehatZ_p(A; t) ∏_a ∈ \textrmFrob_p(A)(1 - at), where \textrmFrob_p(A) := \ a \colon -2√(p) ≤ a ≤ 2√(p) and a ≡ p+1 \pmod|A|\, and \widehatZ_p(A;t) is an easily computed polynomial that is determined by the first \lceil(2\lfloor 2√(p)\rfloor)/(|A|)\rceil power moments. These rational zeta functions have two natural applications. We find rational generating functions in weight aspect for traces of Hecke operators on S_k(Γ) for various congruence subgroups Γ. We also prove congruence relations for power moments by making use of known congruences for traces of Hecke operators.

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• Hypergeometric functions over F_p and relations to elliptic curves and modular forms
• Zeta elements for elliptic curves and applications
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