Q-curves, Hecke characters and some Diophantine equations II

• Pacetti, Ariel ^[2] ; Villagra Torcomian, Lucas ^[1]
  1. [1] Universidad Nacional de Córdoba

     Universidad Nacional de Córdoba

     Argentina

     
  2. [2] Universidad de Aveiro. Department of Mathematics
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 67, Nº 0, 2023, págs. 569-599
• Idioma: inglés
• DOI: 10.5565/publmat6722304
• Enlaces
  • Texto completo
• Resumen

  • In the article [25] a general procedure to study solutions of the equations x4 − dy2 = z p was presented for negative values of d. The purpose of the present article is to extend our previous results to positive values of d. On doing so, we give a description of the extension Q(√d, √e)/Q(√d) (where e is a fundamental unit) needed to prove the existence of a Hecke character over Q(√d) with prescribed local conditions. We also extend some “large image” results due to Ellenberg regarding images of Galois representations coming from Q-curves from imaginary to real quadratic fields.

• Referencias bibliográficas
  • A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m), Math. Ann. 185 (1970), 134–160. DOI: 10.1007/BF01359701
  • W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I: The user language, J. Symbolic Comput. 24(3-4) (1997), 235–265. DOI: 10.1006/jsco.1996.0125
  • D. A. Buell, “Binary Quadratic Forms. Classical Theory and Modern Computations”, Springer-Verlag, New York, 1989. DOI: 10.1007/978-1-4612-4542-1
  • D. Bump, “Automorphic Forms and Representations”, Cambridge Studies in Advanced Mathematics 55, Cambridge University Press, Cambridge, 1997. DOI:...
  • J. H. E. Cohn, The Diophantine equation x 4 − Dy2 = 1, II, Acta Arith. 78(4) (1997), 401–403. DOI: 10.4064/aa-78-4-401-403
  • D. A. Cox, “Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication”, Second edition, Pure and Applied Mathematics...
  • J. Cremona and A. Pacetti, On elliptic curves of prime power conductor over imaginary quadratic fields with class number 1, Proc. Lond. Math....
  • H. B. Daniels and A. Lozano-Robledo ´ , On the number of isomorphism classes of CM elliptic curves defined over a number field, J. Number...
  • H. Darmon and A. Granville, On the equations zm = F(x, y) and Axp +Byq = Czr, Bull. London Math. Soc. 27(6) (1995), 513–543. DOI:...
  • M. Derickx, S. Kamienny, W. Stein, and M. Stoll, Torsion points on elliptic curves over number fields of small degree, Preprint (2017). arXiv:1707.00364
  • J.-M. Deshouillers and F. Dress, Sommes de diviseurs et structure multiplicative des entiers, Acta Arith. 49(4) (1988), 341–375. DOI: 10.4064/aa-49-4-341-375
  • L. Dieulefait and J. Jimenez Urroz ´ , Solving Fermat-type equations via modular Q-curves over polyquadratic fields, J. Reine Angew. Math....
  • J. S. Ellenberg, Galois representations attached to Q-curves and the generalized Fermat equation A4 + B2 = Cp, Amer. J. Math. 126(4)...
  • 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:...
  • N. Freitas and S. Siksek, Criteria for irreducibility of mod p representations of Frey curves, J. Théor. Nombres Bordeaux 27(1) (2015), 67–76....
  • C. F. Gauss, “Disquisitiones arithmeticae”, Translated and with a preface by Arthur A. Clarke, Revised by William C. Waterhouse, Cornelius...
  • C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture (II), Invent. Math. 178(3) (2009), 505–586. DOI: 10.1007/s00222-009-0206-6
  • C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture, in: “Proceedings of the International Congress of Mathematicians”, Vol. II,...
  • A. Kraus, Courbes elliptiques semi-stables sur les corps de nombres, Int. J. Number Theory 3(4) (2007), 611–633. DOI: 10.1142/S1793042107001127
  • S. Le Fourn, Nonvanishing of central values of L-functions of newforms in S2(Γ0(dp2)) twisted by quadratic characters, Canad. Math. Bull....
  • W. Ljunggren, Über die Gleichung x4 −Dy2 = 1, Arch. Math. Naturvid. 45(5) (1942), 61–70.
  • W. Ljunggren, Ein Satz ¨uber die diophantische Gleichung Ax2−By4 = C (C =1, 2, 4), in: “Tolfte Skandinaviska Matematikerkongressen,...
  • P. Michaud-Jacobs, Fermat’s Last Theorem and modular curves over real quadratic fields, Acta Arith. 203(4) (2022), 319–351. DOI: 10.4064/aa210812-2-4
  • F. Najman and G. C. T¸ urcas¸, Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number...
  • A. Pacetti and L. Villagra Torcomian, Q-curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022), 2817–2865. DOI:...
  • L. Pan, The Fontaine–Mazur conjecture in the residually reducible case, J. Amer. Math. Soc. 35(4) (2022), 1031–1169. DOI: 10.1090/jams/991
  • The PARI group, PARI/GP version 2.12.2 (2019). Available at http://pari.math.u-bordeaux.fr/
  • A. Ranum, The group of classes of congruent quadratic integers with respect to a composite ideal modulus, Trans. Amer. Math. Soc. 11(2) (1910),...
  • K. A. Ribet, Lowering the levels of modular representations without multiplicity one, Internat. Math. Res. Notices 1991(2) (1991), 15–19....
  • K. A. Ribet, Abelian varieties over Q and modular forms, in: “Modular Curves and Abelian Varieties”, Progr. Math. 224, Birkhäuser, Basel,...
  • J.-P. Serre, “Local Fields”, Translated from the French by Marvin Jay Greenberg, Graduate Texts in Mathematics 67, Springer-Verlag, New York-Berlin,...
  • J.-P. Serre, Sur les repr´esentations modulaires de degr´e 2 de Gal(Q/Q), Duke Math. J. 54(1) (1987), 179–230. DOI: 10.1215/S0012-7094-87-05413-5
  • P. G. Walsh, Diophantine equations of the form aX4−bY 2 = ±1, in: “Algebraic Number Theory and Diophantine Analysis” (Graz, 1998), de...
  • S. Zuniga Alterman, Explicit averages of square-free supported functions: to the edge of the convolution method, Colloq. Math. 168(1) (2022),...