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Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$

Peter Beelen, Maria Montanucci, Lara Vicino

TL;DR

The article determines the entire Weierstrass semigroup structure and the full automorphism group for the maximal function field Z3 over F_{q^2} with q ≡ 0 (mod 3), where g3 = ⎣(q^2 − q + 4)/6⎦ and Z3 sits as a degree-3 Galois subfield of the Hermitian function field. Building on the Hermitian realization, it develops two families of special functions and three polynomial families to control local expansions, enabling explicit computation of H(P) for all places, including non-rational ones, and a complete classification of Weierstrass semigroups. The automorphism group is shown to be Aut(Z3) = G ≅ S3 ⋊ C2 for q ≥ 9, matching the Hermitian-derived automorphisms, thereby addressing the maximal-genus uniqueness question (for q ≡ 0 mod 3) via semigroup and orbit analyses. These results provide detailed semigroup data across all places and a complete automorphism description, contributing to the broader program of understanding maximal function fields and their potential uniqueness at the third-largest genus.

Abstract

In this article we complete the work started in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known $\mathbb{F}_{q^2}$-maximal function field $Z_3$ having the third largest genus, for $q \equiv 0 \pmod 3$. The cases $q \equiv 2 \pmod 3$ and $q \equiv 1 \pmod 3$ have been in fact analyzed in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], respectively. As in the other two cases, the function field $Z_3$ arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, $Z_3$ has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of $\mathbb{F}_{q^2}$-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, $\mathrm{Aut}(Z_3)$ is exactly the automorphism group inherited from the Hermitian function field, apart from the case $q=3$.

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$

TL;DR

The article determines the entire Weierstrass semigroup structure and the full automorphism group for the maximal function field Z3 over F_{q^2} with q ≡ 0 (mod 3), where g3 = ⎣(q^2 − q + 4)/6⎦ and Z3 sits as a degree-3 Galois subfield of the Hermitian function field. Building on the Hermitian realization, it develops two families of special functions and three polynomial families to control local expansions, enabling explicit computation of H(P) for all places, including non-rational ones, and a complete classification of Weierstrass semigroups. The automorphism group is shown to be Aut(Z3) = G ≅ S3 ⋊ C2 for q ≥ 9, matching the Hermitian-derived automorphisms, thereby addressing the maximal-genus uniqueness question (for q ≡ 0 mod 3) via semigroup and orbit analyses. These results provide detailed semigroup data across all places and a complete automorphism description, contributing to the broader program of understanding maximal function fields and their potential uniqueness at the third-largest genus.

Abstract

In this article we complete the work started in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known -maximal function field having the third largest genus, for . The cases and have been in fact analyzed in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], respectively. As in the other two cases, the function field arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of -rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, is exactly the automorphism group inherited from the Hermitian function field, apart from the case .

Paper Structure

This paper contains 10 sections, 17 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. The function field $Z_3$ and its properties
  4. Computation of $H(P)$ for some special places $P$ of $Z_3$
  5. Three classes of special polynomials
  6. Two classes of special functions
  7. Weierstrass semigroup at the remaining places of $\overline{Z}_3$
  8. Computation of H(P) for the remaining rational places of Z_3
  9. Weierstrass semigroups at the non-F_q2̂-rational places
  10. The automorphism group of Z_3

Key Result

Lemma 2.1

The divisor of the differential $dy$ of $Z_3$ is given by For any function $f \in L((m-1)(q+2)P_\infty)$ and any place $P$ of $\overline{Z}_3$ different from $P_\infty$, the value $v_P(f) + 1$ is a gap of $P$, that is, $v_P(f) + 1 \in G(P)$.

Theorems & Definitions (43)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 33 more