Murmurations of quadratic and cubic characters over function fields

Abstract

We compute the murmuration density for the family of quadratic characters over function fields. We show that the expectation of this density coincides with a correction term arising in the traces of high powers of the Frobenius class in this family, for which we determine large-genus asymptotics. This correction term admits a description in terms of the stable homology of the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point. We further identify this correction as a lower-order term in the one-level density and we obtain a non-vanishing result at the central point for quadratic $L$-functions over function fields, via an alternative approach to using the ratios conjecture. We also compute the murmuration density for the thin family of primitive cubic characters in the Kummer setting.