DISTRIBUTION OF MOMENTS OF HURWITZ CLASS NUMBERS IN ARITHMETIC PROGRESSIONS AND HOLOMORPHIC PROJECTION

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 = m (mod M) and consider the ratio of the 2k-th moment to the zeroeth moment for H(4n - t2) as one varies n. The special case n = pr yields as a consequence asymptotic formulas for moments of the trace t = m (mod M) of Frobenius on elliptic curves over finite fields with pr elements.

Automated summary

In this paper, we investigate the moments of Hurwitz class numbers in relation to fixed arithmetic progressions of imaginary quadratic orders. Specifically, we examine the ratio of the 2k-th moment to the zeroeth moment for H(4n - t2) as n varies, where t is fixed in an arithmetic progression t = m (mod M). As a consequence, we obtain asymptotic formulas for moments of the trace t = m (mod M) of Frobenius on elliptic curves over finite fields with pr elements when n = pr.