Algebra & Number Theory

Le théorème de Fermat sur certains corps de nombres totalement réels

Alain Kraus

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Résumé

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.

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1553565644

Digital Object Identifier
doi:10.2140/ant.2019.13.301

Mathematical Reviews number (MathSciNet)
MR3927048

Zentralblatt MATH identifier
07042061

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

Keywords
Fermat equation number fields elliptic curves modular method

Citation

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. https://projecteuclid.org/euclid.ant/1553565644


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References

  • S. Anni and S. Siksek, “Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$”, Algebra Number Theory 10:6 (2016), 1147–1172.
  • W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system, I: The user language”, J. Symbolic Comput. 24:3-4 (1997), 235–265.
  • C. Breuil, B. Conrad, F. Diamond, and R. Taylor, “On the modularity of elliptic curves over $\mathbb{Q}$: wild 3-adic exercises”, J. Amer. Math. Soc. 14:4 (2001), 843–939.
  • J. Browkin, “The $abc$-conjecture for algebraic numbers”, Acta Math. Sin. $($Engl. Ser.$)$ 22:1 (2006), 211–222.
  • P. Bruin and F. Najman, “A criterion to rule out torsion groups for elliptic curves over number fields”, Res. Number Theory 2 (2016), art. id. 3.
  • J. W. S. Cassels and A. Fröhlich (editors), Algebraic number theory (Brighton, 1965), Academic Press, London, 1967.
  • H. Cohen, Advanced topics in computational number theory, Graduate Texts in Math. 193, Springer, 2000.
  • D. A. Cox, Primes of the form $x^2 + ny^2$, Wiley, New York, 1989.
  • J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge Univ. Press, 1997.
  • J. E. Cremona and L. Dembélé, “Modular forms over number fields”, preprint, 2014, https://tinyurl.com/cremdem.
  • S. R. Dahmen, Classical and modular methods applied to Diophantine equations, Ph.D. thesis, Utrecht University, 2008, https://tinyurl.com/srdahmen.
  • L. Dembélé and J. Voight, “Explicit methods for Hilbert modular forms”, pp. 135–198 in Elliptic curves, Hilbert modular forms and Galois deformations, edited by H. Darmon et al., Birkhäuser, Basel, 2013.
  • M. Derickx, Torsion points on elliptic curves over number fields of small degree, Ph.D. thesis, Leiden University, 2016, http://hdl.handle.net/1887/43186.
  • N. Freitas and S. Siksek, “The asymptotic Fermat's last theorem for five-sixths of real quadratic fields”, Compos. Math. 151:8 (2015), 1395–1415.
  • N. Freitas and S. Siksek, “Fermat's last theorem over some small real quadratic fields”, Algebra Number Theory 9:4 (2015), 875–895.
  • N. Freitas and S. Siksek, “On the asymptotic Fermat's last theorem”, preprint, 2018.
  • N. Freitas, B. V. Le Hung, and S. Siksek, “Elliptic curves over real quadratic fields are modular”, Invent. Math. 201:1 (2015), 159–206.
  • T. Fukuda and K. Komatsu, “Weber's class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, III”, Int. J. Number Theory 7:6 (2011), 1627–1635.
  • B. H. Gross and D. E. Rohrlich, “Some results on the Mordell–Weil group of the Jacobian of the Fermat curve”, Invent. Math. 44:3 (1978), 201–224.
  • F. Jarvis and P. Meekin, “The Fermat equation over ${\mathbb{Q}}(\sqrt{2})$”, J. Number Theory 109:1 (2004), 182–196.
  • M. Klassen and P. Tzermias, “Algebraic points of low degree on the Fermat quintic”, Acta Arith. 82:4 (1997), 393–401.
  • A. Kraus, “Courbes elliptiques semi-stables sur les corps de nombres”, Int. J. Number Theory 3:4 (2007), 611–633.
  • A. Kraus, “Quartic points on the Fermat quintic”, Ann. Math. Blaise Pascal 25:1 (2018), 199–205.
  • LMFDB Collaboration, “The L-functions and modular forms database”, 2013, http://www.lmfdb.org.
  • B. Mazur and J. Vélu, “Courbes de Weil de conducteur $26$”, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), 743–745.
  • L. Merel, “Bornes pour la torsion des courbes elliptiques sur les corps de nombres”, Invent. Math. 124:1-3 (1996), 437–449.
  • J. Oesterlé, note non publiée, 1996.
  • I. Papadopoulos, “Sur la classification de Néron des courbes elliptiques en caractéristique résiduelle $2$ et $3$”, J. Number Theory 44:2 (1993), 119–152.
  • P. Parent, “No $17$-torsion on elliptic curves over cubic number fields”, J. Théor. Nombres Bordeaux 15:3 (2003), 831–838.
  • PARI Group, PARI/GP version 2.7.3, 2015, http://pari.math.u-bordeaux.fr.
  • K. A. Ribet, “On modular representations of ${\rm Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}})$ arising from modular forms”, Invent. Math. 100:2 (1990), 431–476.
  • J. J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Math. 148, Springer, 1995.
  • M. H. Şengün and S. Siksek, “On the asymptotic Fermat's last theorem over number fields”, Comment. Math. Helv. 93:2 (2018), 359–375.
  • J.-P. Serre, “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”, Invent. Math. 15:4 (1972), 259–331.
  • J.-P. Serre, “Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}})$”, Duke Math. J. 54:1 (1987), 179–230.
  • C. L. Siegel, “Über einige Anwendungen diophantischer Approximationen”, Abh. Preuß. Akad. Wiss., Phys.-Math. Kl. 1929:1 (1929), 1–41.
  • J. H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Math. 106, Springer, 2009.
  • N. P. Smart, The algorithmic resolution of Diophantine equations, London Math. Soc. Student Texts 41, Cambridge Univ. Press, 1998.
  • R. Taylor and A. Wiles, “Ring-theoretic properties of certain Hecke algebras”, Ann. of Math. $(2)$ 141:3 (1995), 553–572.
  • J. A. Thorne, “Elliptic curves over $\mathbb{Q}_{\infty}$ are modular”, 2015. À paraître dans J. Eur. Math. Soc.
  • P. Tzermias, “Algebraic points of low degree on the Fermat curve of degree seven”, Manuscripta Math. 97:4 (1998), 483–488.
  • J. Voight, “Tables of totally real number fields”, https://math.dartmouth.edu/~jvoight/nf-tables.
  • W. C. Waterhouse, “Abelian varieties over finite fields”, Ann. Sci. École Norm. Sup. $(4)$ 2:4 (1969), 521–560.
  • A. Wiles, “Modular elliptic curves and Fermat's last theorem”, Ann. of Math. $(2)$ 141:3 (1995), 443–551.