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On Weierstrass semigroups of maximal Fermat function fields

Peter Beelen, Maria Montanucci, Marie Frank vom Braucke

Abstract

In this article we explicitly determine the Weierstrass semigroup at any place of some $\mathbb{F}_{q^2}$-maximal Fermat function fields $\mathcal{F}_m$, namely for $m=(q+1)/2$ and $m=(q+1)/3$. These famous function fields arise as Galois subfields of the Hermitian function field, and even though they have been intensively studied in the literature, the Weierstrass semigroup at every place is still not fully known. Surprisingly enough this problem is in fact quite involved and $\mathcal{F}_m$ has many different types of Weierstrass semigroups. Moreover, its set of Weierstrass places is much richer than its set of rational places.

On Weierstrass semigroups of maximal Fermat function fields

Abstract

In this article we explicitly determine the Weierstrass semigroup at any place of some -maximal Fermat function fields , namely for and . These famous function fields arise as Galois subfields of the Hermitian function field, and even though they have been intensively studied in the literature, the Weierstrass semigroup at every place is still not fully known. Surprisingly enough this problem is in fact quite involved and has many different types of Weierstrass semigroups. Moreover, its set of Weierstrass places is much richer than its set of rational places.
Paper Structure (14 sections, 16 theorems, 124 equations)

This paper contains 14 sections, 16 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The function field $\mathcal{F}_m$.
  4. Weierstrass semigroups and numerical semigroups.
  5. Weierstrass semigroups in $\mathcal{F}_m$, $m \mid (q+1)$
  6. Weierstrass semigroup $H(P)$ for $P\in \mathcal{O}$
  7. Weierstrass semigroup $H(P)$ for $P\in\mathbb{P}_{\mathcal{F}_m}\setminus \mathcal{O}$
  8. The case $m=(q+1)/2$
  9. The case $m=(q+1)/3$
  10. Proof of Theorem \ref{['conj:gaps']}
  11. Construction of gaps
  12. Counting of gaps
  13. Counting of gaps for large $\mathcal{P}$-order
  14. Counting of gaps for small $\mathcal{P}$-order

Key Result

Lemma 2.4

Let $m>3$ be a proper divisor of $q+1$, and let $\zeta_m$ be a primitive $m$-th root of unity. Define where Then we have that $\text{Aut}(\mathcal{F}_m) =G_m$ where $G_m$ denotes the group $G_m:=\langle A_m, H_m\rangle$. In particular $\text{Aut}(\mathcal{F}_m)$ is isomorphic to a semidirect product of an abelian group of order $m^2$ (direct product of two cyclic groups of order $m$) and a symme

Theorems & Definitions (26)

  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.9
  • proof
  • Corollary 3.11
  • proof
  • ...and 16 more