Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection

TL;DR

This paper investigates the distribution of Hurwitz class number moments within arithmetic progressions, deriving asymptotic formulas for moments of Frobenius traces on elliptic curves over finite fields.

Contribution

It introduces a novel analysis of Hurwitz class number moments in arithmetic progressions and connects these to Frobenius trace moments on elliptic curves.

Findings

Derived asymptotic formulas for moments of Frobenius traces

Analyzed distribution of Hurwitz class numbers in arithmetic progressions

Established connections between class numbers and elliptic curve traces

Abstract

In this paper, we study moments of Hurwitz class numbers associated to imaginary quadratic orders restricted into fixed arithmetic progressions. In particular, we fix $t$ in an arithmetic progression $t\equiv m\pmod{M}$ and consider the ratio of the $2k$-th moment to the zeroeth moment for $H(4n-t^2)$ as one varies $n$. The special case $n=p^r$ yields as a consequence asymptotic formulas for moments of the trace $t\equiv m\pmod{M}$ of Frobenius on elliptic curves over finite fields with $p^r$ elements.