A Global Generic Locus for Elementary Abelian p-covers of P^1 with two Ramification Points
Abstract
In this paper, we study elementary abelian p-covers of P1 totally ramified at exactly two points and unramified elsewhere. Such covers are isomorphic to the rank-ℓ Artin--Schreier curves $ X_{f, \ell} : y^{p^\ell} - y = f(x), $ where f ∈ F̅p[x± 1] has a pole of order d1>0 at $0$ and order d2>0 at ∞. After fixing d1 and d2, the cover Xf, ℓ is generic if its Newton polygon is minimal as f varies over all Laurent polynomials with pole orders d1 and d2. For p sufficiently large relative to d1 and d2, we construct a combinatorial cost polygon GL(d1,d2,p) and prove that its attainment is characterized by the nonvanishing of a homogeneous polynomial in the coefficients of f. Moreover, we prove that these conditions stabilize: there exists a Zariski-open dense subset U in the coefficient space that is defined over Q and depends only on d1 and d2, whereby for any f whose coefficients lie in U(Q̅), the reduction is generic at every prime ideal above sufficiently large rational primes, and for every rank ℓ permitted by the residue field. This result extends Zhu's result for one ramification point.
Related
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- Abelian p-extensions with restricted p-ramification and the cyclotomic ℤ2-extension of ℚ (√q)
- Abelian surfaces of small conductor from genus 3 double covers
- Eventual Exactness and Rigidity for Abelian Covers of Groups
- Integer group determinants for the elementary abelian group of order 49
BibTeX
@misc{moore2026global,
title = {{A Global Generic Locus for Elementary Abelian p-covers of P^1 with two Ramification Points}},
author = {Robert Moore and Hui June Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02398},
eprint = {2611.02398},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02398},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Global Generic Locus for Elementary Abelian p-covers of P^1 with two Ramification Points
AU - Robert Moore
AU - Hui June Zhu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02398
UR - https://arxiv.org/abs/2611.02398
AB - In this paper, we study elementary abelian p-covers of P^1 totally ramified at exactly two points and unramified elsewhere. Such covers are isomorphic to the rank-ℓ Artin--Schreier curves $ X_{f, \ell} : y^{p^\ell} - y = f(x), $ where f ∈ F̅_p[x^± 1] has a pole of order d_1>0 at $0$ and order d_2>0 at ∞. After fixing d_1 and d_2, the cover X_f, ℓ is generic if its Newton polygon is minimal as f varies over all Laurent polynomials with pole orders d_1 and d_2. For p sufficiently large relative to d_1 and d_2, we construct a combinatorial cost polygon GL(d_1,d_2,p) and prove that its attainment is characterized by the nonvanishing of a homogeneous polynomial in the coefficients of f. Moreover, we prove that these conditions stabilize: there exists a Zariski-open dense subset U in the coefficient space that is defined over Q and depends only on d_1 and d_2, whereby for any f whose coefficients lie in U(Q̅), the reduction is generic at every prime ideal above sufficiently large rational primes, and for every rank ℓ permitted by the residue field. This result extends Zhu's result for one ramification point.
ER -