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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\ \, \left (\operatorname {mod} \, M \right) 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\ \, \left (\operatorname {mod} \, M \right) of Frobenius on elliptic curves over finite fields with p^r elements. [ABSTRACT FROM AUTHOR]

Given an elliptic curve E defined over \mathbb {C}, let E^{\times } be an open subset of E obtained by removing a point. In this paper, we show that the i-th Betti number of the unordered configuration space \mathrm {Conf}^{n}(E^{\times }) of n points on E^{\times } appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the \mathbb {F}_{q}-point counts of \mathrm {Conf}^{n}(E^{\times }), which can be obtained from the zeta function of E over any fixed finite field \mathbb {F}_{q}. We show that the mixed Hodge structure of the i-th singular cohomology group H^{i}(\mathrm {Conf}^{n}(E^{\times })) with complex coefficients is pure of weight w(i), an explicit integer we provide in this paper. This purity statement implies our main result about the Betti numbers and the Hodge numbers. Our proof uses Totaro's spectral sequence computation that describes the weight filtration of the mixed Hodge structure on H^{i}(\mathrm {Conf}^{n}(E^{\times })). [ABSTRACT FROM AUTHOR]