{"ID":23047519,"CreatedAt":"2026-09-17T05:47:22.378464935Z","UpdatedAt":"2026-09-17T05:47:22.378464935Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19095","arxiv_id":"2609.19095","title":"Explicit equations of Galois subfields of Hermitian function fields with respect to decomposition groups","abstract":"Let $q$ be a prime power and $\\mathbb{F}_{q^2}$ be the finite fields of $q^2$ elements. The Hermitian function field $H=\\mathbb{F}_{q^2}(x,y)$ defined by $y^q+y=x^{q+1}$ is a well-known maximal function field with the largest possible genus. Let $A(P_\\infty)$ be the decomposition group of the infinity place $P_\\infty$ of $H$ which is the common pole of $x$ and $y$. For every subgroup $G\\le A(P_\\infty)$, 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 $\\mathbb{F}_p$-quadratic polynomial on the right-hand side. For $q=27$, we can construct a genus-two subfield $D\\subset H$ that is not isomorphic to $H^J$ for any subgroup $J\\le \\text{Aut}(H)$ 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.","short_abstract":"Let $q$ be a prime power and $\\mathbb{F}_{q^2}$ be the finite fields of $q^2$ elements. The Hermitian function field $H=\\mathbb{F}_{q^2}(x,y)$ defined by $y^q+y=x^{q+1}$ is a well-known maximal function field with the largest possible genus. Let $A(P_\\infty)$ be the decomposition group of the infinity place $P_\\infty$...","url_abs":"https://arxiv.org/abs/2609.19095","url_pdf":"https://arxiv.org/pdf/2609.19095v1","authors":"[\"Liming Ma\",\"Yipeng Wang\"]","published":"2026-09-16T17:29:23Z","proceeding":"math.NT","tasks":"[\"math.NT\"]","methods":"[]","has_code":false}