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On a Galois cover of the Hermitian curve of genus $\mathfrak{g}=\frac{1}{8}(q-1)^2$

Barbara Gatti, Gioia Schulte

Abstract

In the study of algebraic curves with many points over a finite field, a well known general problem is to understanding better the properties of $\mathbb{F}_{q^2}$-maximal curves whose genera fall in the higher part of the spectrum of the genera of all $\mathbb{F}_{q^2}$-maximal curves. This problem is still open for genera smaller than $ \lfloor \frac{1}{6}(q^2-q+4) \rfloor$. In this paper we consider the case of $\mathfrak{g}=\frac{1}{8}(q-1)^2$ where $q\equiv 1\pmod{4}$ and the curve is the Galois cover of the Hermitian curve w.r.t to a cyclic automorphism group of order $4$. Our contributions concern Frobenius embedding, Weierstrass semigroups and automorphism groups.

On a Galois cover of the Hermitian curve of genus $\mathfrak{g}=\frac{1}{8}(q-1)^2$

Abstract

In the study of algebraic curves with many points over a finite field, a well known general problem is to understanding better the properties of -maximal curves whose genera fall in the higher part of the spectrum of the genera of all -maximal curves. This problem is still open for genera smaller than . In this paper we consider the case of where and the curve is the Galois cover of the Hermitian curve w.r.t to a cyclic automorphism group of order . Our contributions concern Frobenius embedding, Weierstrass semigroups and automorphism groups.

Paper Structure

This paper contains 14 sections, 20 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Background on function fields
  3. The Hermitian curve and its function field
  4. Background on algebraic curves
  5. Order sequence
  6. Projections
  7. Bézout's theorem
  8. Embedding of a plane curve in a higher dimension projective space
  9. Natural Embedding
  10. Construction, inherited automorphism group and genus
  11. Branches centered in the singular point
  12. Weierstrass semigroup and natural embedding
  13. The $\mathbb{F}_{q^2}$-rational automorphism group
  14. On the geometry of the natural embedding

Key Result

theorem 1

Let $\mathcal{F}$ the $\mathbb{F}_{q^2}$-maximal curve as in (x). Then

Theorems & Definitions (39)

  • theorem 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • proof
  • lemma 5
  • ...and 29 more