Bilinear forms with Kloosterman sums and applications

We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov range). We then derive applications to the second moment of cusp forms twisted by characters modulo primes,...

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Bibliographic Details
Published in:Annals of mathematics 2017-09, Vol.186 (2), p.413-500
Main Authors: Kowalski, Emmanuel, Michel, Philippe, Sawin, Will
Format: Article
Language:English
Subjects:
Algebra
Automorphisms
Coordinate systems
Fourier transformations
Infinity
Integers
Mathematical theorems
Morphisms
Tensors
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Summary:We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov range). We then derive applications to the second moment of cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on GL₃. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially ℓ-adic cohomology and the Riemann Hypothesis.
ISSN:0003-486X
DOI:10.4007/annals.2017.186.2.2