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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)

Abstract

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
$$ \Delta = q\mathop \prod \limits_1^\infty (1 - q^n )^{24} = \sum\limits_1^\infty {\tau (n)q^n } $$
(1)
and note that the associated Dirichlet series has an Euler product
$$ \sum {\tau (n)n^{ - s} } = \prod (1 - \tau (p)p^{ - s} + p^{11 - 2s} )^{ - 1} $$
(2)
so that all the τ(n) are known as soon as the τ(p) are.

Keywords

Modular Form Galois Group Cusp Form Residue Class Congruence Relation 
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

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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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