Cohen-Lenstra Heuristics and Vanishing of Zeta Functions for Cyclic Covers of Projective Lines over Finite Fields

Ellenberg-Venkatesh-Westerland proved a Cohen-Lenstra result for imaginary quadratic function fields over finite fields by asymptotically counting points on Hurwitz schemes, which parameterize tamely ramified G-covers of the projective line. Moreover, Ellenberg-Li-Shusterman used the methods of Ellenberg-Venkatesh-Westerland to prove that a fixed complex number vanishes on almost no Zeta functions of hyperelliptic curves over finite fields, with respect to a limit taking the genera of the curves to infinity and then the sizes of the base fields to infinity. I will talk about my progress towards extending their results to higher degree function fields and curves.