Algebra & Number Theory

Le théorème de Fermat sur certains corps de nombres totalement réels

Alain Kraus

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Soit K un corps de nombres totalement réel. Pour tout nombre premier p5, notons Fp la courbe de Fermat d’équation xp+yp+zp=0. Sous l’hypothèse que 2 est totalement ramifié dans K, on établit quelques résultats sur l’ensemble Fp(K) des points de Fp rationnels sur K. On obtient un critère pour que le théorème de Fermat asymptotique soit vrai sur K, critère relatif à l’ensemble des newforms modulaires paraboliques de Hilbert sur K, de poids parallèle 2 et de niveau l’idéal premier au-dessus de 2. Il peut souvent se tester simplement numériquement, notamment quand le nombre de classes restreint de K vaut 1. Par ailleurs, en utilisant la méthode modulaire, on démontre le théorème de Fermat de façon effective, sur certains corps de nombres dont les degrés sur sont 3,4,5,6 et 8.


Let K be a totally real number field. For all prime number p5, let us denote by Fp the Fermat curve of equation xp+yp+zp=0. Under the assumption that 2 is totally ramified in K, we establish some results about the set Fp(K) of points of Fp rational over K. We obtain a criterion so that the asymptotic Fermat’s last theorem is true over K, criterion related to the set of Hilbert modular cuspidal newforms over K, of parallel weight 2 and of level the prime ideal above 2. It is often simply testable numerically, particularly if the narrow class number of K is 1. Furthermore, using the modular method, we prove Fermat’s last theorem effectively, over some number fields whose degrees over are 3,4,5,6 and 8.

Article information

Algebra Number Theory, Volume 13, Number 2 (2019), 301-332.

Received: 24 September 2017
Revised: 13 April 2018
Accepted: 23 September 2018
First available in Project Euclid: 26 March 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D41: Higher degree equations; Fermat's equation
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 11R37: Class field theory

Fermat equation number fields elliptic curves modular method


Kraus, Alain. Le théorème de Fermat sur certains corps de nombres totalement réels. Algebra Number Theory 13 (2019), no. 2, 301--332. doi:10.2140/ant.2019.13.301.

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