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

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 $\mathcal{X}⟶\mathcal{Y}$. In this paper we are interested in studying whether the converse of this statement is true, i.e., if there exists a morphism $\mathcal{X}⟶\mathcal{Y}$ 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.