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

TL;DR

This paper extends existing formulas relating traces of elliptic curves over finite fields to Hecke operators, accommodating cases where the subgroup order is divisible by the characteristic p, thus broadening the scope of previous results.

Contribution

It generalizes the trace formulas for elliptic curves with specified subgroups to include cases where the subgroup order is divisible by p, expanding prior work.

Findings

Extended trace formulas to divisible subgroup orders

Unified previous theorems under a broader framework

Provided new computational tools for elliptic curve analysis

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 $\mathrm{SL}_2(\mathbb{Z})$. 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 $p$.