Normal points on Artin–Schreier curves over finite fields
In 2022, S. D. Cohen and the two authors introduced and studied the concept of $(r, n)$-freeness on finite cyclic groups $G$ for suitable integers $r$, $n$, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of $G$. Combining this machinery with some characte...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2025-06-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.740/ |
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| Summary: | In 2022, S. D. Cohen and the two authors introduced and studied the concept of $(r, n)$-freeness on finite cyclic groups $G$ for suitable integers $r$, $n$, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of $G$. Combining this machinery with some character sum techniques, they explored the existence of points $(x_0, y_0)$ on affine curves $y^n=f(x)$ defined over a finite field $\mathbb{F}$ whose coordinates are generators of the multiplicative cyclic group $\mathbb{F}^*$. In this paper we develop the natural additive counterpart of this work for finite fields. Namely, any finite extension $\mathbb{E}$ of a finite field $\mathbb{F}$ with $Q$ elements is a cyclic $\mathbb{F}[x]$-module induced by the Frobenius automorphism $\alpha \mapsto \alpha ^{Q}$, and any generator of this module is said to be a normal element over $\mathbb{F}$. We introduce and study the concept of $(f, g)$-freeness on this module structure for suitable polynomials $f, g\in \mathbb{F}[x]$. As a main application of the machinery developed in this paper, we study the existence of $\mathbb{F}_{p^n}$-rational points in the Artin–Schreier curve $\mathfrak{A}_f : y^p-y=f(x)$ whose coordinates are normal over the prime field $\mathbb{F}_p$ and establish concrete results. |
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| ISSN: | 1778-3569 |