Elliptic curves over a finite field with a specified subgroup and the trace formula

2024/12/24 by Tadahiro Katsuoka, Katsuoka, Tadahiro
Computer Science · Mathematics · #11F72 #11G20 #14G15 #14H52 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2412.18181

openalex publication_date 2024/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Ihara and Birch obtained a formula expressing the sum of powers of the traces of elliptic curves over a fixed finite field of characteristic p in terms of the traces of Hecke operators for SL₂(ℤ). Generalizing the theorems of Ihara and Birch, for a finite abelian group A whose order is coprime to p, Kaplan and Petrow gave a formula for statistical description of powers of the traces of elliptic curves which contain subgroups isomorphic to A. In this paper, we generalize the theorems of Ihara, Birch, and Kaplan--Petrow to the case where the order of A is divisible by p.

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