On the existence of a morphism between certain Artin-Schreier curves
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| Format: | Preprint |
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2026
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| _version_ | 1866917278303387648 |
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| author | Lucas, Beatriz Barbero Lia, Stefano McGuire, Gary |
| author_facet | Lucas, Beatriz Barbero Lia, Stefano McGuire, Gary |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_15717 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the existence of a morphism between certain Artin-Schreier curves Lucas, Beatriz Barbero Lia, Stefano McGuire, Gary Algebraic Geometry 11G20 (Primary), 14H30, 14H37 (Secondary) 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. |
| title | On the existence of a morphism between certain Artin-Schreier curves |
| topic | Algebraic Geometry 11G20 (Primary), 14H30, 14H37 (Secondary) |
| url | https://arxiv.org/abs/2602.15717 |