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 $t\equiv m\pmod{M}$ 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.