Lower Order Terms for Expected Value of Traces of Frobenius of a Family of Cyclic Covers of ${\mathbb{P}}_{{\mathbb{F}}_q}^1$ and One-Level Densities

Abstract

We consider the expected value of $\mbox{Tr}(Θ_C^n)$ where $C$ runs over a thin family of $r$-cyclic covers of ${\mathbb{P}}_{{\mathbb{F}}_q}^1$ for any $r$. We obtain many lower order terms dependent on the divisors of $r$. We use these results to calculate the one-level density of the family and hypothesize a refined one-level density result.

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