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Pré-Publication, Document De Travail Année : 2024

POINTS TOTALEMENT RÉELS DE LA COURBE x^5 + y^5 + z^5 = 0

Résumé

Let $\overline \Q$ be an algebraic closure of $\Q$ and $\Q^{tr}$ be the subfield of $\overline \Q$ obtained by taking the union of all totally real number fields. For any prime $p\geq 3$, let $F_p/\Q$ be the Fermat curve of equation $x^p+y^p+z^p=0$. In 1996, Pop has shown that the field $\Q^{tr}$ is large. In particular, the set $F_p(\Q^{tr})$ of the points of $F_p$ rational over $\Q^{tr}$ is infinite. How to explicit non-trivial points $(xyz\neq 0$) in $F_p(\Q^{tr})$ ? If one has $p\geq 5$, it seems that the only points already known in $F_p(\Q^{tr})$ are those of $F_p(\Q)$ and they are trivial. In this paper, we investigate this question in case $p=5$. There are no totally real fields whose degree over $\Q$ 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 $\Q$.
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hal-04544826 , version 1 (13-04-2024)

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Domaine public

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  • HAL Id : hal-04544826 , version 1

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Alain Kraus. POINTS TOTALEMENT RÉELS DE LA COURBE x^5 + y^5 + z^5 = 0. 2024. ⟨hal-04544826⟩
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