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P-ADIC Properties of Modular Schemes and Modular Forms

  • Nicholas M. Katz
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 350)

Abstract

This expose represents an attempt to understand some of the recent work of Atkin, Swinnerton-Dyer, and Serre on the congruence properties of the q-expansion coefficients of modular forms from the point of view of the theory of moduli of elliptic curves, as developed abstractly by Igusa and recently reconsidered by Deligne. In this optic, a modular form of weight k and level n becomes a section of a certain line bundle \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } ^{ \otimes k} \) on the modular variety Mn which “classifies” elliptic curves with level n structure (the level n structure is introduced for purely technical reasons). The modular variety Mn is a smooth curve over ℤ[l/n], whose “physical appearance” is the same whether we view it over ℂ (where it becomes ϕ(n) copies of the quotient of the upper half plane by the principal congruence subgroup Г(n) of SL(2,ℤ)) or over the algebraic closure of ℤ/pℤ, (by “reduction modulo p”) for primes p not dividing n. This very fact rules out the possibility of obtaining p-adic properties of modular forms simply by studying the geometry of Mn ⊗ℤ/pℤ and its line bundles \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } ^{ \otimes k} \); we can only obtain the reductions modulo p of identical relations which hold over ℂ.

Keywords

Modular Form Elliptic Curve Elliptic Curf Eisenstein Series Newton Polygon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 0.
    Adolphson, A.: Thesis, Princeton University 1973.Google Scholar
  2. 1.
    Atkin, A. O. L.: Congruence Hecke operators, Proc. Symp. Pure Math., vol. 12Google Scholar
  3. 2.
    —: Congruences for modular forms. Proceedings of the IBM Conference on Computers in Mathematical Research, Blaricium, 1966. North-Holland (1967).Google Scholar
  4. 3.
    —, and J. N. O’Brien: Some properties of p(n) and c(n) modulo powers of 13. TAMS 126, (1967), 442–459.CrossRefMathSciNetGoogle Scholar
  5. 4.
    Cartier, P.: Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris 244, (1957), 426–428.zbMATHMathSciNetGoogle Scholar
  6. 5.
    —: Modules associés à un groupe formel commutatif.Courbes typiques. C. R. Acad. Sci. Paris 256, (1967), 129–131.MathSciNetGoogle Scholar
  7. 6.
    —: Groupes formels, course at I.H.E.S., Spring, 1972. (Notes by J. F. Boutot available (?) from I.H.E.S., 91-Bures-sur-Yvette, France.)Google Scholar
  8. 7.
    Deligne, P.: Formes modulaires et représentations ℓ-adiques. Exposé 355. Séminaire N. Bourbaki 1968/1969. Lecture Notes in Mathematics 179, Berlin-Heidelberg-New York: Springer 1969.Google Scholar
  9. 8.
    —: Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Mathematics 163. Berlin-Heidelberg-New York: Springer 1970.zbMATHGoogle Scholar
  10. 9.
    —: Courbes Elliptiques: Formulaire (d’après J. Tate). Multigraph available from I.H.E.S., 91-Bures-sur-Yvette, France, 1968.Google Scholar
  11. 10.
    —, and M. Rapoport: Article in preparation on moduli of elliptic curves.Google Scholar
  12. 11.
    Dwork, B.: P-adic cycles, Pub. Math. I.H.E.S. 37, (1969), 27–115.zbMATHMathSciNetGoogle Scholar
  13. 12.
    —: On Hecke Polynomials, Inventiones Math. 12(1971), 249–256.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 13.
    —: Normalized Period Matrices I, II. Annals of Math. 94, 2nd series, (1971), 337–388, and to appear in Annals of Math.Google Scholar
  15. 14.
    —: Article in this volume.zbMATHMathSciNetGoogle Scholar
  16. 15.
    Grothendieck, A.: Fondements de la Géométrie Algébrique, Secrétariat Mathématique, 11 rue Pierre Curie, Paris 5e, France, 1962.Google Scholar
  17. 15.
    bis —: Formule de Lefschetz et rationalité des fonctions L, Exposé 279, Séminaire Bourbaki 1964/1965.Google Scholar
  18. 16.
    Hasse, H.: Existenz separabler zyklischer unverzweigter Erweiterungskörper vom Primzahlgrade über elliptischen Funktionenkörpern der Charakteristik p. J. Reine angew. Math. 172, (1934), 77–85.Google Scholar
  19. 17.
    Igusa, J.: Class number of a definite quaternion with prime discriminant, Proc. Natl. Acad. Sci. 44, (1958), 312–314.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 18.
    —: Kroneckerian model of fields of elliptic modular functions, Amer. J. Math. 81, (1959), 561–577.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 19.
    —: Fibre systems of Jacobian varieties III, Amer. J. Math. 81, (1959), 453–476.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 20.
    —: On the transformation theory of elliptic functions, Amer. J. Math. 81, (1959), 436–452.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 21.
    —: On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan 20, (1968), 96–106.zbMATHMathSciNetCrossRefGoogle Scholar
  24. 22.
    Ihara, Y.: An invariant multiple differential attached to the field of elliptic modular functions of characteristic p. Amer. J. Math. 78, (1971), 137–147.MathSciNetGoogle Scholar
  25. 23.
    Katz, N.: Une formule de congruence pour la fonction zeta. Exposé 22, SGA 7, 1969, to appear in Springer Lecture Notes in Mathematics. (Preprint available from I.H.E.S., 91-Bures-sur-Yvette, France.)Google Scholar
  26. 24.
    —: Nilpotent connections and the monodromy theorem-applications of a result of Turrittin, Pub. Math. I.H.E.S. 39, (1971), 355–412.Google Scholar
  27. 25.
    —: Travaux de Dwork. Exposé 409, Séminaire N. Bourbaki 1971/72, Springer Lecture Notes in Mathematics, 317, (1973), 167–200.CrossRefGoogle Scholar
  28. 26.
    —: Algebraic solutions of differential equations (p-curvature and the Hodge filtration). Invent. Math. 18, (1972), 1–118.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 27.
    —, and T. Oda: On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8, (1968), 199–213.zbMATHMathSciNetGoogle Scholar
  30. 28.
    Koike, M.: Congruences between modular forms and functions and applications to a conjecture of Atkin, to appear.Google Scholar
  31. 29.
    Lehner, J.: Lectures on modular forms. National Bureau of Standards, Applied Mathematics Series 61, Washington, D.C., 1969.Google Scholar
  32. 30.
    Lubin, J., J.-P. Serre and J. Tate: Elliptic curves and formal groups, Woods Hole Summer Institute 1964 (mimeographed notes).Google Scholar
  33. 31.
    Lubin, J.: One-parameter formal Lie groups over p-adic integer rings, Ann. of Math. 80, 2nd series (1964), 464–484.CrossRefMathSciNetGoogle Scholar
  34. 32.
    —: Finite subgroups and isogenies of one-parameter formal groups, Ann. of Math. 85, 2nd series (1967), 296–302.CrossRefMathSciNetGoogle Scholar
  35. 33.
    —: Newton factorizations of polynomials, to appear.Google Scholar
  36. 33.
    bis —: Canonical subgroups of formal groups, secret notes.Google Scholar
  37. 34.
    Messing, W.: The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lecture Notes in Mathematics 264, Berlin-Heidelberg-New York: Springer 1972.zbMATHGoogle Scholar
  38. 35.
    —: Two functoriality, to appear.Google Scholar
  39. 36.
    Monsky, P.: Formal cohomology III-Trace Formulas. Ann. of Math. 93, 2nd series (1971), 315–343.CrossRefMathSciNetGoogle Scholar
  40. 37.
    Newman, M.: Congruences for the coefficients of modular forms and for the coefficients of j(τ). Proc. A.M.S. 9, (1958), 609–612.zbMATHCrossRefGoogle Scholar
  41. 38.
    Roquette, P.: Analytic theory of elliptic functions over local fields. Göttigen: Vanderhoeck und Ruprecht, 1970.zbMATHGoogle Scholar
  42. 39.
    Serre, J.-P.: Endomorphismes complètement continus des espaces de Banach p-adiques. Pub. Math. I.H.E.S. 12, (1962).Google Scholar
  43. 40.
    —: Course at Collège de France, spring 1972.Google Scholar
  44. 41.
    —: Congruences et formes modulaires. Exposé 416, Séminaire N. Bourbaki, 1971/72, Lecture Notes in Math. 317, (1973), Springer, 319–338.CrossRefMathSciNetGoogle Scholar
  45. 42.
    —: Formes modulaires et fonctions zêta p-adiques, these Proceedings.Google Scholar
  46. 42 1/2.
    —: Cours d’arithmétique. Paris: Presses Univ. de France 1970.Google Scholar
  47. 43.
    Swinnerton-Dyer, H. P. F.: On ℓ-adic representations and congruences for coefficients of modular forms, these Proceedings.Google Scholar
  48. 44.
    Tate, J.: Elliptic curves with bad reduction. Lecture at the 1967 Advanced Science Summer Seminar, Bowdoin College, 1967.Google Scholar
  49. 45.
    —: Rigid analytic spaces. Ihventiones Math. 12, (1971), 257–289.Google Scholar
  50. 46.
    Whittaker, E. T. and G. N. Watson: A course of modern analysis, Cambridge, Cambridge University Press, 1962.zbMATHGoogle Scholar
  51. 47.
    Deligne, P., Cohomologie à Supports Propres, Exposé 17, SGA 4, to appear in Springer Lecture Notes in Mathematics.Google Scholar
  52. 48.
    Roos, J. E., Sur les foncteurs dérivés de \( \underleftarrow {\lim } \). Applications, C. R. Acad. Sci. Paris, tome 252, 1961, pp. 3702–04.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • Nicholas M. Katz
    • 1
  1. 1.Dept. of Math.Univ. of PrincetonPrincetonUSA

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