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 ℓ-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.
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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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DOI: https://doi.org/10.1007/s00039-015-0310-2
Keywords and phrases
- Modular forms
- Fourier coefficients
- Hecke eigenvalues
- Hecke orbits
- horocycles
- ℓ-adic Fourier transform
- Riemann Hypothesis over finite fields