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Algebraic twists of modular forms and Hecke orbits

Abstract

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the -adic Fourier transform introduced by Deligne and studied by Katz and Laumon.

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Correspondence to Philippe Michel.

Additional information

Dedicated to Peter Sarnak on his 61st birthday, with admiration

P. Michel was partially supported by the SNF (grant 200021-137488) and the ERC (Advanced Research Grant 228304). É. Fouvry thanks ETH Zürich, EPF Lausanne and the Institut Universitaire de France for financial support.

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Fouvry, É., Kowalski, E. & Michel, P. Algebraic twists of modular forms and Hecke orbits. Geom. Funct. Anal. 25, 580–657 (2015). https://doi.org/10.1007/s00039-015-0310-2

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Keywords and phrases

  • Modular forms
  • Fourier coefficients
  • Hecke eigenvalues
  • Hecke orbits
  • horocycles
  • -adic Fourier transform
  • Riemann Hypothesis over finite fields

Mathematics Subject Classification

  • 11F11
  • 11F32
  • 11F37
  • 11T23
  • 11L05