On ℓ-ADIC Representations and Congruences for Coefficients of Modular Forms

  • H. P. F. Swinnerton-Dyer
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 350)


The work I shall describe in these lectures has two themes, a classical one going back to Ramanujan [8] and a modern one initiated by Serre [9] and Deligne [3]. To describe the classical theme, let the unique cusp form of weight 12 for the full modular group be written
$$ \Delta = q\mathop \prod \limits_1^\infty (1 - q^n )^{24} = \sum\limits_1^\infty {\tau (n)q^n } $$
and note that the associated Dirichlet series has an Euler product
$$ \sum {\tau (n)n^{ - s} } = \prod (1 - \tau (p)p^{ - s} + p^{11 - 2s} )^{ - 1} $$
so that all the τ(n) are known as soon as the τ(p) are.


Modular Form Galois Group Cusp Form Residue Class Congruence Relation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • H. P. F. Swinnerton-Dyer
    • 1
  1. 1.Dept. of Math.Univ. of CambridgeCambridgeEngland

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