Joint Extension Field Point Counts for Superelliptic Curves
Abstract
Fix a finite field Fq, an integer m≥2 prime to q, and 2≤κ≤ m. For a uniformly chosen κ-power-free polynomial F∈Fq[x] of degree D, we determine the limiting joint distribution of the affine point counts of ym=F(x) over Fq,…,Fqn as D→∞, with q and n fixed. The probability generating function is a finite Euler product with explicit local factors. Equivalently, the limiting vector is a sum of independent closed-point contributions, each of which may affect several extension degrees. We obtain the formula by specializing the equidistribution and motivic Euler product formalism of Bertucci-Howe. We then compute means, covariances, and moment expansions as q varies in a fixed reduced residue class modulo m, after the degree limit has been taken. The one-field specialization recovers the power-free affine point-count distribution of Cheong-Wood-Zaman.
Related
- Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields
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BibTeX
@misc{yao2026joint,
title = {{Joint Extension Field Point Counts for Superelliptic Curves}},
author = {Yuxiang Yao and Xu Zhuang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33847},
eprint = {2610.33847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33847},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Joint Extension Field Point Counts for Superelliptic Curves
AU - Yuxiang Yao
AU - Xu Zhuang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33847
UR - https://arxiv.org/abs/2610.33847
AB - Fix a finite field F_q, an integer m≥2 prime to q, and 2≤κ≤ m. For a uniformly chosen κ-power-free polynomial F∈F_q[x] of degree D, we determine the limiting joint distribution of the affine point counts of y^m=F(x) over F_q,…,F_q^n as D→∞, with q and n fixed. The probability generating function is a finite Euler product with explicit local factors. Equivalently, the limiting vector is a sum of independent closed-point contributions, each of which may affect several extension degrees. We obtain the formula by specializing the equidistribution and motivic Euler product formalism of Bertucci-Howe. We then compute means, covariances, and moment expansions as q varies in a fixed reduced residue class modulo m, after the degree limit has been taken. The one-field specialization recovers the power-free affine point-count distribution of Cheong-Wood-Zaman.
ER -