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)


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.


Modular Form Elliptic Curve Elliptic Curf Eisenstein Series Newton Polygon 
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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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