On the Mertens Conjecture for Elliptic Curves over Finite Fields

TL;DR

This paper explores an analogue of the Mertens conjecture tailored for elliptic curves over finite fields, classifying cases where the conjecture holds based on field size and Frobenius trace.

Contribution

It introduces a new conjecture analogue for elliptic curves over finite fields and classifies the isogeny classes satisfying it using Waterhouse's results.

Findings

Classification of isogeny classes where the conjecture holds

Connection between field size, Frobenius trace, and conjecture validity

Extension of Mertens conjecture concepts to elliptic curves over finite fields

Abstract

We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms the size of the finite field and the trace of the Frobenius endomorphism acting on the curve.