Odd moments for the trace of Frobenius and the Sato--Tate conjecture in arithmetic progressions

TL;DR

This paper investigates the distribution of Frobenius traces of elliptic curves in arithmetic progressions, establishing asymptotic behaviors and proving equidistribution with respect to the Sato--Tate measure.

Contribution

It provides new asymptotic formulas for moments of Frobenius traces restricted to arithmetic progressions and confirms their equidistribution under the Sato--Tate measure.

Findings

Asymptotic behavior of odd moments of Frobenius traces determined.

Distribution of Frobenius traces in arithmetic progressions is Sato--Tate distributed.

Results connect moments of Frobenius traces with Hurwitz class numbers.

Abstract

In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the $(2k+1)$-th moment to the zeroeth moment as the size of the finite field $\mathbb{F}_{p^r}$ goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato--Tate measure.