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

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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.

Many of the results described in these lectures were first obtained in correspondence between Serre and me during the last five years ; the disentanglement of our respective contributions is left to the reader, as an exercise in stylistic analysis. The dedication is from both of us.

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References

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© 1973 Springer-Verlag Berlin Heidelberg

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Swinnerton-Dyer, H.P.F. (1973). On ℓ-ADIC Representations and Congruences for Coefficients of 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_1

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  • DOI: https://doi.org/10.1007/978-3-540-37802-0_1

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