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In this paper, we consider the moments of the trace of Frobenius of ellipticcurves if the trace is restricted to a fixed arithmetic progression. Wedetermine the asymptotic behavior for the ratio of the $(2k+1)$-th moment tothe zeroeth moment as the size of the finite field $\mathbb{F}_{p^r}$ goes toinfinity. These results follow from similar asymptotic formulas relating sumsand moments of Hurwitz class numbers where the sums are restricted to certainarithmetic progressions. As an application, we prove that the distribution ofthe trace of Frobenius in arithmetic progressions is equidistributed withrespect to the Sato--Tate measure.

Odd moments for the trace of Frobenius and the Sato--Tate conjecture in arithmetic progressions