Semistable Abelian Varieties with Small Division Fields

  • Armand Brumer
  • Kenneth Kramer
Part of the Developments in Mathematics book series (DEVM, volume 11)

Abstract

The conjecture of Shimura-Taniyama-Weil, now proved through the work of Wiles and disciples, is only part of the Langlands program. Based on a comparison of the local factors ([And], [Seri]), it also predicts that the L-series of an abelian surface defined over ℚ should be the L-series of a Hecke eigen cusp form of weight 2 on a suitable group commensurable with Sp4 (ℤ). The only decisive examples are related to lifts of automorphic representations of proper subgroups of Sp4, for example the beautiful work of Yoshida ([Yos], [BSP]).

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References

  1. [And]
    A. N. Andrianov, “Quadratic forms and Hecke operators,” Grundlehren der Math., 286. Springer-Verlag, 1987.Google Scholar
  2. [BSP]
    I. S. Böcherer and R. Schulze-Pillot, Siegel modular forms and theta series attached to quaternion algebras, Nagoya Math. J.121 (1991), 35–96.MathSciNetMATHGoogle Scholar
  3. [BK1]
    A. Brumer and K. Kramer, The rank of elliptic curves, Duke Math. J., 44 (1977), 715–743.MathSciNetMATHCrossRefGoogle Scholar
  4. [BK2]
    A. Brumer and K. Kramer, Non-existence of certain semistable abelian varieties, Manuscripta Math. 106 (2001), 291–304.MathSciNetMATHCrossRefGoogle Scholar
  5. [CG]
    J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), 129–174.Google Scholar
  6. [Ed]
    B. Edixhoven, On the prime to p-part of the group of connected components of Néron models, Compositio Math. 97 (1995), 29–49.Google Scholar
  7. [Fa]
    G. Faltings, Endlichkeitssätze für abelsche Varietäten ìber Zahlkörpern, Invent. Math. 73 (1983), 349–366.MathSciNetMATHCrossRefGoogle Scholar
  8. [Fo]
    J.-M. Fontaine, Il n’y a pas de variété abélienne sur ℤ, Invent. Math. 81 (1985), 515–538.Google Scholar
  9. [Gro]
    A. Grothendieck, Modèles de Néron et monodromie, in “Groupes de monodromie en géométrie algébrique. I,” Lecture Notes in Mathematics, 288. Springer-Verlag, New York, 1973.Google Scholar
  10. [Gru]
    F. Grunewald, Some facts from the theory of group schemes,in “Rational Points (G. Faltings and G. Wéstholz, ed.),” Aspects of Mathematics, E6. F. Vieweg & Sohn, Braunschweig, 1984.Google Scholar
  11. [HM]
    F. Hajir and C. Maire, Extensions of number fields with wild ramification of bounded depth, Inter. Math. Res. Notices (2002), 667–696.Google Scholar
  12. [Has]
    K.-I. Hashimoto, The dimension of the spaces of cusp forms on Siegel upper half-plane of degree two, I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 403–488.MathSciNetMATHGoogle Scholar
  13. [Ibu]
    T. Ibukiyama, On relations of dimension of automorphic forms of Sp(2, ℝ) and its compact twist Sp(2,), I, Adv. Studies in Pure Math. 7 (1985), 7–29.MathSciNetGoogle Scholar
  14. [MM]
    J. M. Masley and H. L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Ang. Math. 286/287 (1976), 248–256.MathSciNetGoogle Scholar
  15. [Ma]
    B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S. 47 (1976), 33–186.Google Scholar
  16. [Mc]
    W. McCallum, Duality theorems for Néron models, Duke Math. J. 53 (1986), 1093–1124.MathSciNetMATHCrossRefGoogle Scholar
  17. [MO]
    J.-F. Mestre and J. Oesterlé, Courbes de Weil semi-stables de discriminant une puissance m-iéme, J. Reine Ang. Math. 400 (1989), 171–184.Google Scholar
  18. [Mi]
    I. Miyawaki, Elliptic curves of prime power conductor with ℚ-rational points of finite order, Osaka Math J., 10 (1973), 309–323.MathSciNetMATHGoogle Scholar
  19. [Ne]
    Neumann, Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. II, Math. Nach. 56 (1973), 269–280.MATHCrossRefGoogle Scholar
  20. [Sch]
    N. Schappacher, Tate’s conjecture on the endomorphisms of abelian varieties, in “Rational Points (G. Faltings and G. Wéstholz, ed.),” Aspects of Mathematics, E6. F. Vieweg & Sohn, Braunschweig, 1984.Google Scholar
  21. [Scho]
    R. Schoof, Semistable abelian varieties over ℚ, (2001), preprint.Google Scholar
  22. [Ser1]
    J.-P. Serre, Facteurs locaux des fonctions zéta des variétés algébriques, Sém. DPP, exposé 19 (1969–1970).Google Scholar
  23. [Ser2]
    J.-P. Serre, “Local Fields,” Lecture Noti’s in Mathematics, 67. Springer-Verlag, New York, 1979.Google Scholar
  24. [Ser3]
    J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.MathSciNetMATHCrossRefGoogle Scholar
  25. [Ser4]
    J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(\(\mathop \mathbb{Q}\limits^ - /\mathbb{Q}\)), Duke Math. J. 54 (1987), 179–230.MathSciNetMATHCrossRefGoogle Scholar
  26. [Ser5]
    J.-P. Serre, Oeuvres. Collected papers. IV (1986), 1–55, Springer-Verlag, 2000.Google Scholar
  27. [Set]
    C. B. Setzer, Elliptic curves of prime conductor, J. Lond. Math. Soc. 10 (1975), 367–378.MathSciNetMATHCrossRefGoogle Scholar
  28. [SS]
    Schinzel, A. and Sierpir¨ªski, W., Sur certaines hypothéses concernant les nombres premiers, Acta Arithmetica 4 (1958), 185–208.MATHGoogle Scholar
  29. [Web]
    H.-J. Weber, Hyperelliptic simple factors of Jo(N) with dimension at least 3, Exper. Math. J. 6 (1997), 273–287.MATHGoogle Scholar
  30. [Yos]
    H. Yoshida, On Siegel modular forms obtained from theta seri• J. Reine Ang. Math. 352 (1984), 184–219.MATHCrossRefGoogle Scholar
  31. [Zar]
    Yu. G. Zarhin, A finiteness theorem for unpolarized abelian varieties over number fields with prescribed places of bad reduction, Invent. Math. 79 (1985), 309–321.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Armand Brumer
    • 1
  • Kenneth Kramer
    • 2
  1. 1.Department of MathematicsFordham UniversityBronxUSA
  2. 2.Department of MathematicsQueens College (CUNY)FlushingUSA

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