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Modular Curves as Shimura Variety

  • Haruzo Hida
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

Modular varieties classifying abelian varieties with a level structure, given endomorphism subring and polarization, by making the level structure finer and finer, give rise to a tower of varieties. This tower has a strikingly big automorphism group, which can be described by an adele group of an algebraic group. Such a tower with a big adelic automorphism group is called a Shimura variety (of PEL type; see (PAF)) because Shimura was the first to realize that the symmetry given by well-described automorphisms is essential in studying arithmetic problems and automorphic forms defined over the variety (see (Sh4) and his earlier papers quoted there, including (Sh2)).

References

  1. [ACS]
    K.-W. Lan, Arithmetic Compactifications of PEL-Type Shimura Varieties. London Mathematical Society Monographs, vol. 36 (Princeton University Press, Princeton, NJ, 2013)Google Scholar
  2. [BCM]
    N. Bourbaki, Algèbre Commutative (Hermann, Paris, 1961–1998)Google Scholar
  3. [CRT]
    H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, New York, 1986)Google Scholar
  4. [CSM]
    C.-L. Chai, Compactification of Siegel Moduli Schemes. LMS Lecture Note Series, vol. 107 (Cambridge University Press, New York, 1985)Google Scholar
  5. [DAV]
    G. Faltings, C.-L. Chai, Degeneration of Abelian Varieties (Springer-Verlag, New York, 1990)MATHCrossRefGoogle Scholar
  6. [ECH]
    J.S. Milne, Étale Cohomology (Princeton University Press, Princeton, NJ, 1980)MATHGoogle Scholar
  7. [EDM]
    G. Shimura, Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Springer, New York, 2007)Google Scholar
  8. [FAN]
    D. Ramakrishnan, R.J. Valenza, Fourier Analysis on Number Fields. Graduate Texts in Mathematics, vol. 186 (Springer-Verlag, New York, 1999)Google Scholar
  9. [GME]
    H. Hida, Geometric Modular Forms and Elliptic Curves, 2nd edn. (World Scientific, Singapore, 2011)CrossRefGoogle Scholar
  10. [IAT]
    G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, NJ, 1971)MATHGoogle Scholar
  11. [LAG]
    J.E. Humphreys, Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21 (Springer-Verlag, New York, 1987)Google Scholar
  12. [LFE]
    H. Hida, Elementary Theory of L-Functions and Eisenstein Series. London Mathematical Society Student Texts, vol. 26 (Cambridge University Press, Cambridge, 1993)Google Scholar
  13. [MFM]
    T. Miyake, Modular Forms (Springer-Verlag, New York, 1989)MATHGoogle Scholar
  14. [NMD]
    S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models (Springer-Verlag, New York, 1990)MATHCrossRefGoogle Scholar
  15. [PAF]
    H. Hida, p-Adic Automorphic Forms on Shimura Varieties. Springer Monographs in Mathematics (Springer, New York, 2004)Google Scholar
  16. [Br1]
    M. Brakočević, Anticyclotomic p-adic L-function of central critical Rankin–Selberg L-value. Int. Math. Res. Not. 2011(21), 4967–5018 (2011)MATHGoogle Scholar
  17. [Ch2]
    C.-L. Chai, Monodromy of Hecke-invariant subvarieties. Pure Appl. Math. Q. 1(Special Issue: In memory of Armand Borel), 291–303 (2005)Google Scholar
  18. [Ch4]
    C.-L. Chai, Methods for p-adic monodromy. J. Inst. Math. Jussieu 7, 247–268 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. [D3]
    P. Deligne, Travaux de Shimura, Sem. Bourbaki, Exp. 389. Lect. Notes Math. 244, 123–165 (1971)Google Scholar
  20. [D5]
    P. Deligne, Variétés de Shimura: Interprétation modulaire, et techniques de construction de modéles canoniques. Proc. Symp. Pure Math. 33.2, 247–290 (1979)Google Scholar
  21. [DRa]
    P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques. Lect. Notes Math. 349, 143–316 (1973)MathSciNetCrossRefGoogle Scholar
  22. [DRi]
    P. Deligne, K.A. Ribet, Values of abelian L-functions at negative integers over totally real fields. Invent. Math. 59, 227–286 (1980)MathSciNetMATHCrossRefGoogle Scholar
  23. [H85]
    H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I. Invent. Math. 79, 159–195 (1985)MathSciNetMATHCrossRefGoogle Scholar
  24. [H86b]
    H. Hida, Galois representations into \(GL_{2}(\mathbb{Z}_{p}[[X]])\) attached to ordinary cusp forms. Invent. Math. 85, 545–613 (1986)MathSciNetMATHCrossRefGoogle Scholar
  25. [H88b]
    H. Hida, On p-adic Hecke algebras for GL 2 over totally real fields. Ann. Math. 128, 295–384 (1988)MathSciNetMATHCrossRefGoogle Scholar
  26. [H06b]
    H. Hida, Automorphism groups of Shimura varieties of PEL type. Doc. Math. 11, 25–56 (2006)MathSciNetMATHGoogle Scholar
  27. [H09a]
    H. Hida, Irreducibility of the Igusa tower. Acta Math. Sin. Engl. Ser. 25, 1–20 (2009)MathSciNetMATHCrossRefGoogle Scholar
  28. [I]
    J. Igusa, Kroneckerian model of fields of elliptic modular functions. Am. J. Math. 81, 561–577 (1959)MathSciNetMATHCrossRefGoogle Scholar
  29. [K1]
    N.M. Katz, Serre–Tate local moduli, in Surfaces Algébriques. Lecture Notes in Mathematics, vol. 868 (Springer-Verlag, Berlin, 1978), pp. 138–202Google Scholar
  30. [Ko]
    R. Kottwitz, Points on Shimura varieties over finite fields. J. Am. Math. Soc. 5, 373–444 (1992)MathSciNetMATHCrossRefGoogle Scholar
  31. [Ri1]
    K.A. Ribet, P-adic interpolation via Hilbert modular forms. Proc. Symp. Pure Math. 29, 581–592 (1975)MathSciNetCrossRefGoogle Scholar
  32. [Sh2]
    G. Shimura, On analytic families of polarized abelian varieties and automorphic functions. Ann. Math. 78, 149–192 (1963) ([63b] in [CPS] I)Google Scholar
  33. [Sh3]
    G. Shimura, Moduli and fibre system of abelian varieties. Ann. Math. 83, 294–338 (1966) ([66b] in [CPS] I)Google Scholar
  34. [Sh4]
    G. Shimura, On canonical models of arithmetic quotients of bounded symmetric domains. Ann. Math. 91, 144–222 (1970); II, 92, 528–549 (1970) ([70a–b] in [CPS] II)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Haruzo Hida
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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