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arXiv · 2610.16112

Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields

Hyun Jong Kim
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Abstract

We prove a Cohen-Lenstra type result commenting on the distribution of class group structures amongst function fields of superelliptic curves over finite fields. We also prove a result commenting on the sparsity of such superelliptic curves whose zeta functions vanish at a fixed complex number. These results are proven via point-counting techniques on certain Hurwitz schemes generalizing techniques of Ellenberg-Venkatesh-Westerland and Ellenberg-Li-Shusterman. To obtain these point counts, We prove a unitary big monodromy theorem generalizing big monodromy theorems of Yu and Achter-Pries by appealing to an arithmeticity theorem of Venkataramana.

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BibTeXRIS

Hyun Jong Kim. 2026-10-12. Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields. https://arxiv.org/abs/2610.16112

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