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Double Artin-Schreier extensions of rational function fields with many lifted automorphisms

Herivelto Borges, Jonathan Niemann, Giovanni Zini

Abstract

In this paper we investigate algebraic function fields in positive characteristic mainly obtained as double Artin-Schreier extensions of rational function fields with a plane model. The goal is to extend to such extensions large automorphism groups of the rational function field. In this way, we construct some new families of ordinary function fields and determine their full automorphism groups. Such groups are large with respect to the genus, compared with the known upper bounds on the size of the automorphism group of an ordinary function field.

Double Artin-Schreier extensions of rational function fields with many lifted automorphisms

Abstract

In this paper we investigate algebraic function fields in positive characteristic mainly obtained as double Artin-Schreier extensions of rational function fields with a plane model. The goal is to extend to such extensions large automorphism groups of the rational function field. In this way, we construct some new families of ordinary function fields and determine their full automorphism groups. Such groups are large with respect to the genus, compared with the known upper bounds on the size of the automorphism group of an ordinary function field.
Paper Structure (11 sections, 14 theorems, 107 equations)

This paper contains 11 sections, 14 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Background and preliminary results
  3. Double Artin-Schreier extensions over conics
  4. Three ordinary function fields with many automorphisms
  5. The Zieve function field
  6. The modified Zieve function field
  7. The extended Zieve function field
  8. Full automorphism groups of selected curves
  9. Automorphisms of the Singer function field
  10. Automorphisms of the Zieve function field
  11. Automorphisms of the extended Zieve function field

Key Result

Lemma 2.1

Suppose $u$ and $v$ have the same unique pole in $K(u,v)$, and denote this pole by $P_\infty$. If $\lbrack F : K(u,v) \rbrack = q^2$, then $P_\infty$ is totally ramified in $F/K(u,v)$, it is the only ramified place in $F/K(u,v)$, and the $p$-rank of $F$ is $0$.

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Example 3.2: Artin-Mumford
  • Example 3.3: One rational and one non-rational point at infinity
  • Example 3.4: Singer
  • Remark 3.5
  • ...and 31 more