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

@misc{kim2026cohenlenstra,
  title = {{Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields}},
  author = {Hyun Jong Kim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16112},
  eprint = {2610.16112},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16112},
}

TY  - UNPB
TI  - Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields
AU  - Hyun Jong Kim
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16112
UR  - https://arxiv.org/abs/2610.16112
AB  - 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.
ER  -