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

TL;DR

This paper investigates the conditions under which a morphism exists between certain Artin-Schreier curves, specifically examining if the divisibility condition on parameters is necessary for the existence of such morphisms.

Contribution

It proves that the divisibility condition is necessary for the existence of morphisms between specific Artin-Schreier curves under certain hypotheses.

Findings

The divisibility of parameters is necessary for morphism existence in the studied cases.

The paper establishes conditions under which the converse of a known implication holds.

Both Galois and non-Galois morphisms are considered in the analysis.

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.