Mathematische Zeitschrift

, Volume 280, Issue 3–4, pp 1015–1029 | Cite as

Quadratic \({\mathbb {Q}}\)-curves, units and Hecke \(L\)-values

Article

Abstract

We show that if \(K\) is a quadratic field, and if there exists a quadratic \({\mathbb {Q}}\)-curve \(E/K\) of prime degree \(N\), satisfying weak conditions, then any unit \(u\) of \(O_K\) satisfies a congruence \(u^r\equiv 1\pmod {N}\), where \(r={\mathrm {g.c.d.}}(N-1,12)\). If \(K\) is imaginary quadratic, we prove a congruence, modulo a divisor of \(N\), between an algebraic Hecke character \(\tilde{\psi }\) and, roughly speaking, the elliptic curve. We show that this divisor then occurs in a critical value \(L(\tilde{\psi },2)\), by constructing a non-zero element in a Selmer group and applying a theorem of Kato.

Keywords

\({\mathbb {Q}}\)-curve Real quadratic units Hecke \(L\)-function 

Mathematics Subject Classification

11G05 11R11 11G40 

References

  1. 1.
    Berwick, W.E.H.: Modular invariants expressible in terms of quadratic and cubic irrationalities. Proc. Lond. Math. Soc. 28, 53–69 (1927)MathSciNetGoogle Scholar
  2. 2.
    Bloch, S., Kato, K.: \(L\)-Functions and Tamagawa Numbers of Motives. The Grothendieck Festschrift Volume I, pp. 333–400, Progress in Mathematics, 86, Birkhäuser, Boston (1990)Google Scholar
  3. 3.
    Brown, A.F., Ghate, E.P.: Dihedral congruence primes and class fields of real quadratic fields. J. Number Theory 95, 14–37 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brown, J.: Saito–Kurokawa lifts and applications to the Bloch–Kato conjecture. Compos. Math. 143, 290–322 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Casselman, W.: On Abelian varieties with many endomorphisms and a conjecture of Shimura’s. Invent. Math. 12, 225–236 (1971)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cox, D.A.: Primes of the form \(x^2+ny^2\). Wiley, New York (1989)Google Scholar
  7. 7.
    Deligne, P.: Valeurs de Fonctions \(L\) et Périodes d’Intégrales. In: AMS Proceedings of Symposium on Pure Mathematicals, Vol. 33, part 2, pp. 313–346 (1979)Google Scholar
  8. 8.
    Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular Functions of One Variable, II (Proceedings of International Summer School, University of Antwerp, Antwerp, 1972), pp. 143–316. Lect. Notes in Mathematical, 349. Springer, Berlin (1973)Google Scholar
  9. 9.
    Dembélé, L., Kumar, A.: Examples of Abelian Surfaces with Everywhere Good Reduction. https://homepages.warwick.ac.uk/staff/L.Dembele/paper.html
  10. 10.
    Diamond, F., Flach, M., Guo, L.: The Tamagawa number conjecture of adjoint motives of modular forms. Ann. Sci. École Norm. Sup. (4) 37, 663–727 (2004)MathSciNetMATHGoogle Scholar
  11. 11.
    Elkies, N.: On Elliptic \(K\)-curves, 81–91 in Modular Curves and Abelian Varieties. In: Cremona, J., Lario, J.-C., Quer, J., Ribet, K. (eds.) Progress in Mathematics, Vol. 224, Birkhäuser, Basel (2004)Google Scholar
  12. 12.
    Faltings, G.: Finiteness Theorems for Abelian Varieties over Number Fields. In: Cornell, G., Silverman, J. (eds.) Arithmetic Geometry (Storrs, Conn., 1984), pp. 9–27. Springer, New York (1986)Google Scholar
  13. 13.
    Galbraith, S.: Rational points on \(X_0^+(p)\). Exp. Math. 8, 311–318 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    González, J.: On cubic factors of \(j\)-invariants of quadratic \(\mathbb{Q}\)-curves of prime degree. J. Number Theory 128, 377–389 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    González, J., Lario, J.-C.: Rational and elliptic parametrizations of \(\mathbb{Q}\)-curves. J. Number Thoery 72, 13–31 (1998)CrossRefMATHGoogle Scholar
  16. 16.
    Golyshev, V.: Classification Problems and Mirror Duality. Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics, pp. 88–121, London Mathematical Society Lecture Note Series, 338. Cambridge University Press, Cambridge (2007)Google Scholar
  17. 17.
    Hasegawa, Y.: \(\mathbb{Q}\)-curves over quadratic fields. Manuscr. Math. 94, 347–364 (1997)CrossRefMATHGoogle Scholar
  18. 18.
    Hida, H.: Global quadratic units and Hecke algebras. Doc. Math. 3, 273–284 (1998)MathSciNetMATHGoogle Scholar
  19. 19.
    Hida, H.: Geometric modular forms and elliptic curves. World Scientific, Singapore (2000)CrossRefMATHGoogle Scholar
  20. 20.
    Kato, K.: \(p\)-Adic Hodge theory and values of zeta functions of modular forms. Cohomologies \(p\)-Adiques et Applications Arithmétiques. III. Astérisque No. 295, ix, 117290 (2004)Google Scholar
  21. 21.
    Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture (I). Invent. Math. 178, 485–504 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Koike, M.: Congruences between CUSP forms and linear representations of the Galois group. Nagoya Math. J. 64, 63–85 (1976)MathSciNetGoogle Scholar
  23. 23.
    Ohta, M.: The Representation of Galois Group Attached to Certain Finite Group Schemes, and its Application to Shimura’s Theory, pp. 149–156 in Algebraic Number Theory. Proceeding of International Symposium Kyoto 1976, Japan Society Promotion Science, Tokyo (1977)Google Scholar
  24. 24.
    Quer, J.: \(\mathbb{Q}\)-curves and abelian varieties of \(\text{ GL }_2\) type. Proc. Lond. Math. Soc. (3) 81, 285–317 (2000)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Raynaud, M.: Schémas en groupes de type \((p,\ldots, p)\). Bull. Soc. Math. France 102, 241–280 (1974)MathSciNetGoogle Scholar
  26. 26.
    Ribet, K.: A modular construction of unramified \(p\)-extensions of \(\mathbb{Q}(\mu _p)\). Invent. Math. 34, 151–162 (1976)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ribet, K.: Abelian Varieties over \(\mathbb{Q}\) and Modular Forms. In: Cremona, J., Lario, J.-C., Quer, J., Ribet, K. (eds.) Modular Curves and Abelian Varieties. Progress in Mathematics, Vol. 224, pp. 241–261. Birkhäuser, Basel (2004)Google Scholar
  28. 28.
    Rubin, K.: The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103, 25–68 (1991)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Schappacher, N.: Periods of Hecke Characters. Lecture Notes in Mathematics, 1301. Springer, Berlin (1988)Google Scholar
  30. 30.
    Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259–331 (1972)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math. 88, 492–517 (1968)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton (1971)MATHGoogle Scholar
  33. 33.
    Stein, W. A.: Explicit approaches to modular abelian varieties, Ph. D. thesis. University of California at Berkeley (2000)Google Scholar
  34. 34.
    Yamauchi, T.: The modularity of \(\mathbb{Q}\) curves of degree \(43\). Houston J. Math. 34, 1025–1035 (2008)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK
  2. 2.Algebra and Number Theory SectorInstitute for Information Transmission ProblemsMoscowRussia

Personalised recommendations