Bilinear forms with Kloosterman sums and applications

Emmanuel Kowalski, Philippe Michel, Will Sawin

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)413-500
Number of pages88
JournalAnnals of Mathematics
Issue number2
StatePublished - 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Arithmetic functions in arithmetic progressions
  • Kloosterman sheaves
  • Kloosterman sums
  • Moments of L-functions
  • Monodromy
  • Riemann Hypothesis over finite fields
  • Short exponential sums


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