Explicit equations of Galois subfields of Hermitian function fields with respect to decomposition groups
Abstract
Let q be a prime power and 𝔽q2 be the finite fields of q2 elements. The Hermitian function field H=𝔽q2(x,y) defined by yq+y=xq+1 is a well-known maximal function field with the largest possible genus. Let A(P∞) be the decomposition group of the infinity place P∞ of H which is the common pole of x and y. For every subgroup G≤ A(P∞), we construct explicit generators of Galois subfield HG of H with respect to G and determine an absolutely irreducible equation defining the smooth affine plane model for such a Galois subfield. For p-subgroups, the generators can be chosen so that the defining equation has an additive polynomial on the left-hand side and an 𝔽p-quadratic polynomial on the right-hand side. For q=27, we can construct a genus-two subfield D⊂ H that is not isomorphic to HJ for any subgroup J≤ Aut(H) from the explicit equations of Galois subfields of the Hermitian function field. To the best of our knowledge, this is the first example of a maximal function field covered but not Galois-covered by the same Hermitian function field.
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BibTeX
@misc{ma2026explicit,
title = {{Explicit equations of Galois subfields of Hermitian function fields with respect to decomposition groups}},
author = {Liming Ma and Yipeng Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19095},
eprint = {2609.19095},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19095},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Explicit equations of Galois subfields of Hermitian function fields with respect to decomposition groups
AU - Liming Ma
AU - Yipeng Wang
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19095
UR - https://arxiv.org/abs/2609.19095
AB - Let q be a prime power and 𝔽_q^2 be the finite fields of q^2 elements. The Hermitian function field H=𝔽_q^2(x,y) defined by y^q+y=x^q+1 is a well-known maximal function field with the largest possible genus. Let A(P_∞) be the decomposition group of the infinity place P_∞ of H which is the common pole of x and y. For every subgroup G≤ A(P_∞), we construct explicit generators of Galois subfield H^G of H with respect to G and determine an absolutely irreducible equation defining the smooth affine plane model for such a Galois subfield. For p-subgroups, the generators can be chosen so that the defining equation has an additive polynomial on the left-hand side and an 𝔽_p-quadratic polynomial on the right-hand side. For q=27, we can construct a genus-two subfield D⊂ H that is not isomorphic to H^J for any subgroup J≤ Aut(H) from the explicit equations of Galois subfields of the Hermitian function field. To the best of our knowledge, this is the first example of a maximal function field covered but not Galois-covered by the same Hermitian function field.
ER -