Skip to main content

P-ADIC Properties of Modular Schemes and Modular Forms

  • 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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adolphson, A.: Thesis, Princeton University 1973.

    Google Scholar 

  2. Atkin, A. O. L.: Congruence Hecke operators, Proc. Symp. Pure Math., vol. 12

    Google Scholar 

  3. —: Congruences for modular forms. Proceedings of the IBM Conference on Computers in Mathematical Research, Blaricium, 1966. North-Holland (1967).

    Google Scholar 

  4. —, and J. N. O’Brien: Some properties of p(n) and c(n) modulo powers of 13. TAMS 126, (1967), 442–459.

    CrossRef  MathSciNet  Google Scholar 

  5. Cartier, P.: Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris 244, (1957), 426–428.

    MATH  MathSciNet  Google Scholar 

  6. —: Modules associés à un groupe formel commutatif.Courbes typiques. C. R. Acad. Sci. Paris 256, (1967), 129–131.

    MathSciNet  Google Scholar 

  7. —: 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. 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. —: Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Mathematics 163. Berlin-Heidelberg-New York: Springer 1970.

    MATH  Google Scholar 

  10. —: Courbes Elliptiques: Formulaire (d’après J. Tate). Multigraph available from I.H.E.S., 91-Bures-sur-Yvette, France, 1968.

    Google Scholar 

  11. —, and M. Rapoport: Article in preparation on moduli of elliptic curves.

    Google Scholar 

  12. Dwork, B.: P-adic cycles, Pub. Math. I.H.E.S. 37, (1969), 27–115.

    MATH  MathSciNet  Google Scholar 

  13. —: On Hecke Polynomials, Inventiones Math. 12(1971), 249–256.

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. —: Normalized Period Matrices I, II. Annals of Math. 94, 2nd series, (1971), 337–388, and to appear in Annals of Math.

    Google Scholar 

  15. —: Article in this volume.

    MATH  MathSciNet  Google Scholar 

  16. Grothendieck, A.: Fondements de la Géométrie Algébrique, Secrétariat Mathématique, 11 rue Pierre Curie, Paris 5e, France, 1962.

    Google Scholar 

  17. bis —: Formule de Lefschetz et rationalité des fonctions L, Exposé 279, Séminaire Bourbaki 1964/1965.

    Google Scholar 

  18. 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. Igusa, J.: Class number of a definite quaternion with prime discriminant, Proc. Natl. Acad. Sci. 44, (1958), 312–314.

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. —: Kroneckerian model of fields of elliptic modular functions, Amer. J. Math. 81, (1959), 561–577.

    CrossRef  MATH  MathSciNet  Google Scholar 

  21. —: Fibre systems of Jacobian varieties III, Amer. J. Math. 81, (1959), 453–476.

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. —: On the transformation theory of elliptic functions, Amer. J. Math. 81, (1959), 436–452.

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. —: On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan 20, (1968), 96–106.

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. Ihara, Y.: An invariant multiple differential attached to the field of elliptic modular functions of characteristic p. Amer. J. Math. 78, (1971), 137–147.

    MathSciNet  Google Scholar 

  25. 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. —: 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. —: Travaux de Dwork. Exposé 409, Séminaire N. Bourbaki 1971/72, Springer Lecture Notes in Mathematics, 317, (1973), 167–200.

    CrossRef  Google Scholar 

  28. —: Algebraic solutions of differential equations (p-curvature and the Hodge filtration). Invent. Math. 18, (1972), 1–118.

    CrossRef  MATH  MathSciNet  Google Scholar 

  29. —, and T. Oda: On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8, (1968), 199–213.

    MATH  MathSciNet  Google Scholar 

  30. Koike, M.: Congruences between modular forms and functions and applications to a conjecture of Atkin, to appear.

    Google Scholar 

  31. Lehner, J.: Lectures on modular forms. National Bureau of Standards, Applied Mathematics Series 61, Washington, D.C., 1969.

    Google Scholar 

  32. Lubin, J., J.-P. Serre and J. Tate: Elliptic curves and formal groups, Woods Hole Summer Institute 1964 (mimeographed notes).

    Google Scholar 

  33. Lubin, J.: One-parameter formal Lie groups over p-adic integer rings, Ann. of Math. 80, 2nd series (1964), 464–484.

    CrossRef  MathSciNet  Google Scholar 

  34. —: Finite subgroups and isogenies of one-parameter formal groups, Ann. of Math. 85, 2nd series (1967), 296–302.

    CrossRef  MathSciNet  Google Scholar 

  35. —: Newton factorizations of polynomials, to appear.

    Google Scholar 

  36. bis —: Canonical subgroups of formal groups, secret notes.

    Google Scholar 

  37. 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.

    MATH  Google Scholar 

  38. —: Two functoriality, to appear.

    Google Scholar 

  39. Monsky, P.: Formal cohomology III-Trace Formulas. Ann. of Math. 93, 2nd series (1971), 315–343.

    CrossRef  MathSciNet  Google Scholar 

  40. Newman, M.: Congruences for the coefficients of modular forms and for the coefficients of j(τ). Proc. A.M.S. 9, (1958), 609–612.

    CrossRef  MATH  Google Scholar 

  41. Roquette, P.: Analytic theory of elliptic functions over local fields. Göttigen: Vanderhoeck und Ruprecht, 1970.

    MATH  Google Scholar 

  42. Serre, J.-P.: Endomorphismes complètement continus des espaces de Banach p-adiques. Pub. Math. I.H.E.S. 12, (1962).

    Google Scholar 

  43. —: Course at Collège de France, spring 1972.

    Google Scholar 

  44. —: Congruences et formes modulaires. Exposé 416, Séminaire N. Bourbaki, 1971/72, Lecture Notes in Math. 317, (1973), Springer, 319–338.

    CrossRef  MathSciNet  Google Scholar 

  45. —: Formes modulaires et fonctions zêta p-adiques, these Proceedings.

    Google Scholar 

  46. —: Cours d’arithmétique. Paris: Presses Univ. de France 1970.

    Google Scholar 

  47. Swinnerton-Dyer, H. P. F.: On ℓ-adic representations and congruences for coefficients of modular forms, these Proceedings.

    Google Scholar 

  48. Tate, J.: Elliptic curves with bad reduction. Lecture at the 1967 Advanced Science Summer Seminar, Bowdoin College, 1967.

    Google Scholar 

  49. —: Rigid analytic spaces. Ihventiones Math. 12, (1971), 257–289.

    Google Scholar 

  50. Whittaker, E. T. and G. N. Watson: A course of modern analysis, Cambridge, Cambridge University Press, 1962.

    MATH  Google Scholar 

  51. Deligne, P., Cohomologie à Supports Propres, Exposé 17, SGA 4, to appear in Springer Lecture Notes in Mathematics.

    Google Scholar 

  52. Roos, J. E., Sur les foncteurs dérivés de \( \underleftarrow {\lim } \). Applications, C. R. Acad. Sci. Paris, tome 252, 1961, pp. 3702–04.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1973 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Katz, N.M. (1973). P-ADIC Properties of Modular Schemes and Modular Forms. In: Kuijk, W., Serre, JP. (eds) Modular Functions of One Variable III. Lecture Notes in Mathematics, vol 350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-37802-0_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-37802-0_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06483-1

  • Online ISBN: 978-3-540-37802-0

  • eBook Packages: Springer Book Archive