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On the existence of a morphism between certain Artin-Schreier curves

Beatriz Barbero Lucas, Stefano Lia, Gary McGuire

Abstract

It is well known that, given two curves $\mathcal{X}: y^p+cy=x^m$ and $\mathcal{Y}:y^p+cy=x^n$, defined over $\F_p$, if $n$ divides $m$ then there exists a nonconstant morphism $\mathcal{X} \longrightarrow \mathcal{Y}$. In this paper we are interested in studying whether the converse of this statement is true, i.e., if there exists a morphism $\mathcal{X} \longrightarrow\mathcal{Y}$ then must it be true that $n$ divides $m$? In particular, we consider the case when $m=p^{k}+1$ and $n=p^\ell+1$. We prove that the converse is true under certain hypotheses. We deal with both the cases of Galois morphisms and non-Galois morphisms.

On the existence of a morphism between certain Artin-Schreier curves

Abstract

It is well known that, given two curves and , defined over , if divides then there exists a nonconstant morphism . In this paper we are interested in studying whether the converse of this statement is true, i.e., if there exists a morphism then must it be true that divides ? In particular, we consider the case when and . We prove that the converse is true under certain hypotheses. We deal with both the cases of Galois morphisms and non-Galois morphisms.
Paper Structure (9 sections, 20 theorems, 67 equations)

This paper contains 9 sections, 20 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Relevant notions
  4. The curves $B_k$
  5. Non-Galois Case
  6. About Galois morphisms
  7. Genus of the quotient group
  8. Conditions for $B_{\ell}$ to be a quotient of $B_k$
  9. Results

Key Result

Theorem 1.1

Let $p$ be an odd prime. Let $\ell, k$ be positive integers with $1<\ell <k$. Let $K=\overline{{\mathbb{F}}_p}$. Let $c \in K$ be nonzero. Let $B_k$ be the curve $y^p+c y=x^{p^k+1}$ and let $B_\ell$ be the curve $y^p+c y=x^{p^\ell +1}$. Let $P_k$ and $P_\ell$ be the points at infinity of $B_k$ and $

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Lemma 3.1
  • ...and 25 more