Heilbronn Number Theory: Billinear forms with Kloosterman sums

24 May 2017, 4.00 PM - 24 May 2017, 5.00 PM

Igor Shparlinski

Title: Billinear forms with Kloosterman sums

Abstract: We start with a general introduction to the area which is nowadays known as Klostermania, where the main goal is showing nontrivial cancellations between Kloosterman sums in some families.

We outline some new bounds on bilinear sums with Kloosterman sums and also with some similar sums. In particular, these bounds improve some recent results of V. Blomer, E. Fouvry, E. Kowalski, Ph. Michel and G. Milicevic (2014-2016). As a result we improve the error term in the asymptotic formula for mixed moments of L-series associated with Hecke eigenforms.

We also discuss further extensions of this method (jointly with Tianping Zhang and Kui Liu), which improve some recent results of R. Nunes (2016).

Finally, we outline some possible arithmetic applications of these bounds.

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Tom Oliver, Dan Fretwell

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