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

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 $t\equiv m(\mathrm{mod}M)$ 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 $t\equiv m(\mathrm{mod}M)$ of Frobenius on elliptic curves over finite fields with $p^r$ elements.