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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.08922 |
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| _version_ | 1866929313288290304 |
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| author | Kraus, Alain |
| author_facet | Kraus, Alain |
| contents | Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ and $\mathbb{Q}^{tr}$ be the subfield of $\overline{\mathbb{Q}}$ obtained by taking the union of all totally real number fields. For any prime $p\geq 3$, let $F_p/\mathbb{Q}$ be the Fermat curve of equation $x^p+y^p+z^p=0$. In 1996, Pop has shown that the field $\mathbb{Q}^{tr}$ is large. In particular, the set $F_p(\mathbb{Q}^{tr})$ of the points of $F_p$ rational over $\mathbb{Q}^{tr}$ is infinite. How to explicit non-trivial points $(xyz\neq 0$) in $F_p(\mathbb{Q}^{tr})$ ? If one has $p\geq 5$, it seems that the only points already known in $F_p(\mathbb{Q}^{tr})$ are those of $F_p(\mathbb{Q})$ and they are trivial. In this paper, we investigate this question in case $p=5$. There are no totally real fields whose degree over $\mathbb{Q}$ is at most $5$ over which $F_5$ has non-trivial points. We propose here to explicit infinitely many points of $F_5$ rational over totally real fields of degree $6$ over $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Points totalement réels de la courbe $x^5+y^5+z^5=0$ Kraus, Alain Number Theory Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ and $\mathbb{Q}^{tr}$ be the subfield of $\overline{\mathbb{Q}}$ obtained by taking the union of all totally real number fields. For any prime $p\geq 3$, let $F_p/\mathbb{Q}$ be the Fermat curve of equation $x^p+y^p+z^p=0$. In 1996, Pop has shown that the field $\mathbb{Q}^{tr}$ is large. In particular, the set $F_p(\mathbb{Q}^{tr})$ of the points of $F_p$ rational over $\mathbb{Q}^{tr}$ is infinite. How to explicit non-trivial points $(xyz\neq 0$) in $F_p(\mathbb{Q}^{tr})$ ? If one has $p\geq 5$, it seems that the only points already known in $F_p(\mathbb{Q}^{tr})$ are those of $F_p(\mathbb{Q})$ and they are trivial. In this paper, we investigate this question in case $p=5$. There are no totally real fields whose degree over $\mathbb{Q}$ is at most $5$ over which $F_5$ has non-trivial points. We propose here to explicit infinitely many points of $F_5$ rational over totally real fields of degree $6$ over $\mathbb{Q}$. |
| title | Points totalement réels de la courbe $x^5+y^5+z^5=0$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.08922 |