The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions

TL;DR

This paper investigates the bias in second moments of elliptic curves over finite fields with traces in arithmetic progressions, revealing both positive and negative biases depending on the progression, and provides explicit formulas for related moments.

Contribution

It introduces a new perspective on the bias conjecture for elliptic curves over finite fields in arithmetic progressions and derives explicit formulas for moments of traces and Hurwitz class numbers.

Findings

Bias is positive for a positive density of progressions

Bias is negative for a positive density of progressions

Explicit formulas for moments of Frobenius traces and Hurwitz class numbers

Abstract

In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in an arbitrary fixed arithmetic progression. Contrary to the classical setting of reductions of one-parameter families over the rationals, where it is conjectured by Steven J. Miller that the bias is always negative, we prove that in our setting the bias is positive for a positive density of arithmetic progressions and negative for a positive density of arithmetic progressions. Along the way, we obtain explicit formulas for moments of traces of Frobenius of elliptic curves over finite fields in arithmetic progressions and related moments of Hurwitz class numbers in arithmetic progressions, the distribution of which are of independent interest.