Geometric and Functional Analysis

, Volume 25, Issue 2, pp 580–657 | Cite as

Algebraic twists of modular forms and Hecke orbits

  • Étienne Fouvry
  • Emmanuel Kowalski
  • Philippe MichelEmail author
Open Access


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.

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 


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Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Étienne Fouvry
    • 1
  • Emmanuel Kowalski
    • 2
  • Philippe Michel
    • 3
    Email author
  1. 1.Laboratoire de MathématiqueUniversité Paris SudOrsay CedexFrance
  2. 2.ETH Zürich – D-MATHZürichSwitzerland
  3. 3.EPFL Chaire TANLausanneSwitzerland

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