Ir al contenido

Documat

Some extensions of the modular method and Fermat equations of signature (13,13,n)

  • Billerey, Nicolas [5] ; Chen, Imin [1] ; Dembélé, Lassina [2] ; Dieulefait, Luis [3] ; Freitas, Nuno [4]
    1. [1] Simon Fraser University

      Simon Fraser University

      Canadá

    2. [2] University of Luxembourg

      University of Luxembourg

      Luxemburgo

    3. [3] Universitat de Barcelona

      Universitat de Barcelona

      Barcelona, España

    4. [4] Instituto de Ciencias Matemáticas

      Instituto de Ciencias Matemáticas

      Madrid, España

    5. [5] Université Clermont Auvergne
    Mostrar afiliaciones +
  • Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 67, Nº 0, 2023, págs. 715-741
  • Idioma: inglés
  • DOI: 10.5565/publmat6722309
  • Enlaces
  • Resumen

    • We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer n ≥ 2, the equation x13 + y 13 = 3zn has no non-trivial primitive solutions. In particular, we present four elimination techniques which are based on: (1) establishing reducibility of certain residual Galois representations over a totally real field; (2) generalizing image of inertia arguments to the setting of abelian surfaces; (3) establishing congruences of Hilbert modular forms without the use of often impractical Sturm bounds; and (4) a unit sieve argument which combines information from classical descent and the modular method. The extensions are of broader applicability and provide further evidence that it is possible to obtain a complete resolution of a family of generalized Fermat equations by remaining within the framework of the modular method. As a further illustration of this, we complete a theorem of Anni–Siksek to show that, for `, m ≥ 5, the only primitive solutions to the equation x 2` + y 2m = z 13 are trivial.

  • Referencias bibliográficas
    • S. Anni and S. Siksek, Modular elliptic curves over real abelian fields and the generalized Fermat equation x2` + y 2m = z p,...
    • M. A. Bennett and C. M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math. 56(1) (2004),...
    • N. Billerey, I. Chen, L. Dembel´ e, L. Dieulefait, and N. Freitas ´ , Supporting Magma program files for this paper, available as ancillary...
    • N. Billerey, I. Chen, L. Dieulefait, and N. Freitas, A multi-Frey approach to Fermat equations of signature (r, r, p), Trans. Amer. Math....
    • W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language, Computational algebra and number theory (London, 1993),...
    • C. Breuil and F. Diamond, Formes modulaires de Hilbert modulo p et valeurs d’extensions entre caract`eres galoisiens, Ann. Sci. Ec. Norm....
    • Y. Bugeaud, M. Mignotte, and S. Siksek, A multi-Frey approach to some multi-parameter families of Diophantine equations, Canad. J. Math. 60(3)...
    • J. I. Burgos Gil and A. Pacetti, Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields, Math. Comp. 86(306) (2017),...
    • C. J. Bushnell and G. Henniart, “The Local Langlands Conjecture for GL(2)”, Grundlehren der mathematischen Wissenschaften 335, Springer-Verlag,...
    • E. Cali, Defaut de semi-stabilite des courbes elliptiques dans le cas non ramifie, Canad. J. Math. 56(4) (2004), 673–698. DOI: 10.4153/CJM-2004-031-6
    • H. Carayol, Sur les repr´esentations l-adiques associ´ees aux formes modulaires de Hilbert, Ann. Sci. Ecole Norm. Sup. (4) ´ 19(3) (1986),...
    • I. Chen and A. Koutsianas, A modular approach to Fermat equations of signature (p, p, 5) using Frey hyperelliptic curves, Preprint (2022)....
    • E. Costa, N. Mascot, J. Sijsling, and J. Voight, Rigorous computation of the endomorphism ring of a Jacobian, Math. Comp. 88(317) (2019),...
    • S. R. Dahmen and S. Siksek, Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164(1) (2014), 65–100. DOI: 10.4064/aa164-1-5
    • H. Darmon, Rigid local systems, Hilbert modular forms, and Fermat’s last theorem, Duke Math. J. 102(3) (2000), 413–449. DOI: 10.1215/S0012-7094-00-10233-5
    • L. Dembele, An intriguing hyperelliptic Shimura curve quotient of genus 16, Algebra Number Theory 14(10) (2020), 2713–2742. DOI: 10.2140/ant.2020.14.2713
    • L. Dembel´ e and A. Kumar ´ , Examples of abelian surfaces with everywhere good reduction, Math. Ann. 364(3-4) (2016), 1365–1392. DOI: 10.1007/s00208-015-1252-6
    • L. Dembele and J. Voight , Explicit methods for Hilbert modular forms, in: “Elliptic Curves, Hilbert Modular Forms and Galois Deformations”,...
    • F. Diamond, An extension of Wiles’ results, in: “Modular Forms and Fermat’s Last Theorem” (Boston, MA, 1995), Springer, New York, 1997, pp....
    • T. Dokchitser and C. Doris, 3-torsion and conductor of genus 2 curves, Math. Comp. 88(318) (2019), 1913–1927. DOI: 10.1090/mcom/3387
    • C. Doris, Package for computing the conductor or genus 2 curves. Available at https://cjdoris.github.io/Genus2Conductor/#installation.
    • N. Elkies and A. Kumar, K3 surfaces and equations for Hilbert modular surfaces, Algebra Number Theory 8(10) (2014), 2297–2411. DOI: 10.2140/ant.2014.8.2297
    • J. S. Ellenberg, Serre’s conjecture over F9, Ann. of Math. (2) 161(3) (2005), 1111–1142. DOI: 10.4007/annals.2005.161.1111
    • W. Feit, “The Representation Theory of Finite Groups”, North-Holland Mathematical Library 25, North-Holland Publishing Co., Amsterdam-New...
    • N. Freitas, On the Fermat-type equation x3 + y3 = zp, Comment. Math. Helv. 91(2) (2016), 295–304. DOI: 10.4171/CMH/386
    • N. Freitas, B. V. Le Hung, and S. Siksek, Elliptic curves over real quadratic fields are modular, Invent. Math. 201(1) (2015), 159–206. DOI:...
    • M. Greenberg and J. Voight, Computing systems of Hecke eigenvalues associated to Hilbert modular forms, Math. Comp. 80(274) (2011), 1071–1092. DOI:...
    • C. Khare and J.-P. Wintenberger, On Serre’s conjecture for 2-dimensional mod p representations of Gal(Q/Q), Ann. of Math. (2) 169(1) (2009),...
    • M. Kisin, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170(3) (2009), 1085–1180. DOI: 10.4007/annals.2009.170.1085
    • A. Kraus, Sur le defaut de semi-stabilite des courbes elliptiques a reduction additive, Manuscripta Math. 69(4) (1990), 353–385. DOI: 10.1007/BF02567933
    • A. Kraus, Sur l’´equation a3 + b3 = cp, Experiment. Math. 7(1) (1998), 1–13. DOI: 10.1080/10586458.1998.10504355
    • P. Kutzko, The Langlands conjecture for Gl2 of a local field, Ann. of Math. (2) 112(2) (1980), 381–412. DOI: 10.2307/1971151
    • The LMFDB Collaboration, The L-functions and modular forms database (2013). http://www.lmfdb.org.
    • Q. Liu, Courbes stables de genre 2 et leur sch´ema de modules, Math. Ann. 295(2) (1993), 201–222. DOI: 10.1007/BF01444884
    • K. Martin, The Jacquet–Langlands correspondence, Eisenstein congruences, and integral L-values in weight 2, Math. Res. Lett. 24(6) (2017),...
    • J.-F. Mestre, Construction de courbes de genre 2 `a partir de leurs modules, in: “Effective Methods in Algebraic Geometry” (Castiglioncello,...
    • J. Nekova´rˇ, Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two, Canad. J. Math. 64(3) (2012), 588–668....
    • K. A. Ribet, Abelian varieties over Q and modular forms, in: “Modular Curves and Abelian Varieties”, Progr. Math. 224, Birkh¨auser, Basel,...
    • K. A. Ribet, Non-optimal levels of mod ` reducible Galois representations or Modularity of residually reducible representations, CRM Lecture...
    • J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88(3) (1968), 492–517. DOI: 10.2307/1970722
    • G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45(3) (1978), 637–679. DOI: 10.1215/S0012-7094-78-04529-5
    • A. Shnidman, Quadratic twists of abelian varieties with real multiplication, Int. Math. Res. Not. IMRN 2021(5) (2021), 3267–3298. DOI: 10.1093/imrn/rnz185
    • J. Wilson, Curves of genus 2 with real multiplication by a square root of 5, Thesis (Ph.D.)-University of Oxford (1998).
    • H. Yoo, Non-optimal levels of a reducible mod ` modular representation, Trans. Amer. Math. Soc. 371(6) (2019), 3805–3830. DOI: 10.1090/tran/7314
    • S. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153(1) (2001), 27–147. DOI: 10.2307/2661372
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno