Elliptic and Modular Curves over Rings

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


In Chap. 1, we studied nonvanishing properties modulo p of Dirichlet L-values, and in Sect. 3.5, we outlined the proof of the vanishing of the μ-invariant of p-adic Hecke L-functions.


  1. [ABV]
    D. Mumford, Abelian Varieties. TIFR Studies in Mathematics (Oxford University Press, New York, 1994)Google Scholar
  2. [ACM]
    G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions (Princeton University Press, Princeton, NJ, 1998)zbMATHGoogle Scholar
  3. [ALG]
    R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer-Verlag, New York, 1977)Google Scholar
  4. [AME]
    N.M. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves. Annals of Mathematics Studies, vol. 108 (Princeton University Press, Princeton, NJ, 1985)Google Scholar
  5. [BCM]
    N. Bourbaki, Algèbre Commutative (Hermann, Paris, 1961–1998)Google Scholar
  6. [CBT]
    W. Messing, The Crystals Associated to Barsotti–Tate Groups; with Applications to Abelian Schemes. Lecture Notes in Mathematics, vol. 264 (Springer-Verlag, New York, 1972)Google Scholar
  7. [CPS]
    G. Shimura, Collected Papers, vols. I, II, III, IV (Springer, New York, 2002)CrossRefGoogle Scholar
  8. [CRT]
    H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, New York, 1986)Google Scholar
  9. [EDM]
    G. Shimura, Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Springer, New York, 2007)Google Scholar
  10. [EGA]
    A. Grothendieck, J. Dieudonné, Eléments de Géométrie Algébrique. Publications IHES, vol. 4 (1960), vol. 8 (1961), vol. 11 (1961), vol. 17 (1963), vol. 20 (1964), vol. 24 (1965), vol. 28 (1966), vol. 32 (1967)Google Scholar
  11. [GIT]
    D. Mumford, Geometric Invariant Theory. Ergebnisse, vol. 34 (Springer-Verlag, New York, 1965)Google Scholar
  12. [GME]
    H. Hida, Geometric Modular Forms and Elliptic Curves, 2nd edn. (World Scientific, Singapore, 2011)CrossRefGoogle Scholar
  13. [IAT]
    G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, NJ, 1971)zbMATHGoogle Scholar
  14. [LAG]
    J.E. Humphreys, Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21 (Springer-Verlag, New York, 1987)Google Scholar
  15. [NMD]
    S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models (Springer-Verlag, New York, 1990)zbMATHCrossRefGoogle Scholar
  16. [RAG]
    J.C. Jantzen, Representations of Algebraic Groups (Academic Press, Orlando, FL, 1987)zbMATHGoogle Scholar
  17. [Br1]
    M. Brakočević, Anticyclotomic p-adic L-function of central critical Rankin–Selberg L-value. Int. Math. Res. Not. 2011(21), 4967–5018 (2011)zbMATHGoogle Scholar
  18. [Ch1]
    C.-L. Chai, Families of ordinary abelian varieties: canonical coordinates, p-adic monodromy, Tate-linear subvarieties and Hecke orbits, preprint 2003 (posted at:
  19. [I]
    J. Igusa, Kroneckerian model of fields of elliptic modular functions. Am. J. Math. 81, 561–577 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  20. [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
  21. [Mo]
    B. Moonen, Serre–Tate theory for moduli spaces of PEL type. Ann. Sci. Éc. Norm. Sup. (4) 37, 223–269 (2004)Google Scholar
  22. [ST]
    J.-P. Serre, J. Tate, Good reduction of abelian varieties. Ann. Math. 88, 452–517 (1968) (Serre’s Œubres II 472–497, No. 79)Google Scholar
  23. [T1]
    J. Tate, p-Divisible groups, in Proceedings of a Conference on Local Fields, Driebergen, 1966 (Springer-Verlag, Berlin, 1967), pp. 158–183Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

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

Personalised recommendations