Geodesic
Bilinear forms with Kloosterman sums and applications
Emmanuel Kowalski1 ; Philippe Michel2 ; Will Sawin3
1 ETH Zürich, Departement Mathematik, Zürich, Switzerland, Rämistrasse 101<br/>CH-8092 Zürich, Switzerland
2 Ecole Polytechnique Fédérale de Lausanne, Institute of Mathematics, EPFL, Station 8, CH-1015 Lausanne, Switzerland
3 ETH Zürich, Institute for Theoretical Studies, CH-8092 Zürich, Switzerland
Annals of mathematics, Tome 186 (2017) no. 2, pp. 413-500.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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 $\mathrm{GL}_3$. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.
DOI : 10.4007/annals.2017.186.2.2

Emmanuel Kowalski 1 ; Philippe Michel 2 ; Will Sawin 3

1 ETH Zürich, Departement Mathematik, Zürich, Switzerland, Rämistrasse 101<br/>CH-8092 Zürich, Switzerland
2 Ecole Polytechnique Fédérale de Lausanne, Institute of Mathematics, EPFL, Station 8, CH-1015 Lausanne, Switzerland
3 ETH Zürich, Institute for Theoretical Studies, CH-8092 Zürich, Switzerland 
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     title = {Bilinear forms with {Kloosterman} sums and applications},
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Emmanuel Kowalski; Philippe Michel; Will Sawin. Bilinear forms with Kloosterman sums and applications. Annals of mathematics, Tome 186 (2017) no. 2, pp. 413-500. doi : 10.4007/annals.2017.186.2.2. http://geodesic.mathdoc.fr/articles/10.4007/annals.2017.186.2.2/

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