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Review of Scheme Theory

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

Abstract

Elliptic curves are algebraic groups in addition to being algebraic curves; a modular form classifies elliptic curves by assigning a value in a ring A to an elliptic curve with some extra structures defined over A. From a different viewpoint, automorphic forms are generally defined on the adele points of a linear algebraic group; in particular, modular forms are functions on \(GL_{2}(\mathbb{A})\) (e.g., (AAG)). Thus, a minimal amount of knowledge of group schemes is necessary to describe elliptic curves and modular forms. Group schemes are best understood from the functorial viewpoint of scheme theory, regarding a scheme as a functor from the category of algebras to sets taking each algebra R to the set of R-rational points S(R) of the scheme S.

References

  1. [AAG]
    S.S. Gelbart, Automorphic Forms on Adele Groups. Annals of Mathematics Studies, vol. 83 (Princeton University Press, Princeton, NJ, 1975)Google Scholar
  2. [ABV]
    D. Mumford, Abelian Varieties. TIFR Studies in Mathematics (Oxford University Press, New York, 1994)Google Scholar
  3. [ALG]
    R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer-Verlag, New York, 1977)Google Scholar
  4. [BCM]
    N. Bourbaki, Algèbre Commutative (Hermann, Paris, 1961–1998)Google Scholar
  5. [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
  6. [CMA]
    H. Matsumura, Commutative Algebra (Benjamin, New York, 1970)Google Scholar
  7. [CRT]
    H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, New York, 1986)Google Scholar
  8. [ECH]
    J.S. Milne, Étale Cohomology (Princeton University Press, Princeton, NJ, 1980)Google Scholar
  9. [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
  10. [FGA]
    A. Grothendieck, Fondements de la Géométrie Algébrique. Séminaire Bourbaki exp. no. 149 (1956/57), 182 (1958/59), 190 (1959/60), 195 (1959/60), 212 (1960/61), 221 (1960/61), 232 (1961/62) (Benjamin, New York, 1966)Google Scholar
  11. [GME]
    H. Hida, Geometric Modular Forms and Elliptic Curves, 2nd edn. (World Scientific, Singapore, 2011)Google Scholar
  12. [IAT]
    G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, NJ, 1971)Google Scholar
  13. [MFG]
    H. Hida, Modular Forms and Galois Cohomology. Cambridge Studies in Advanced Mathematics, vol. 69 (Cambridge University Press, Cambridge, 2000)Google Scholar
  14. [NMD]
    S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models (Springer-Verlag, New York, 1990)Google Scholar
  15. [PAF]
    H. Hida, p-Adic Automorphic Forms on Shimura Varieties. Springer Monographs in Mathematics (Springer, New York, 2004)Google Scholar
  16. [RAG]
    J.C. Jantzen, Representations of Algebraic Groups (Academic Press, Orlando, FL, 1987)Google Scholar
  17. [SGA]
    A. Grothendieck, Revetements Étale et Groupe Fondamental. Séminaire de geometrie algébrique. Lecture Notes in Mathematics, vol. 224 (Springer-Verlag, Berlin, 1971)Google Scholar
  18. [TCF]
    M. Nagata, Theory of Commutative Fields (American Mathematical Society, Providence, RI, 1993)Google Scholar
  19. [Ch1]
    C.-L. Chai, Families of ordinary abelian varieties: canonical coordinates, p-adic monodromy, Tate-linear subvarieties and Hecke orbits, preprint 2003 (posted at: www.math.upenn.edu/~chai)
  20. [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

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