**Date/Time**

Date(s) - 16/11/2023*9:30 am - 10:30 am*

Time and location: 9:30-10:20am, HH207.

Speaker: David Nguyen (Queen’s University)

Title: An improved upper bound on a class of exponential sums

Abstract: In Yitang Zhang’s proof of bounded gaps between primes (2014), a certain 3-variable Kloosterman sum played a crucial role and was one of the deepest parts of his proof. Further improvements on bounded gaps will likely require deeper understanding of this sum. This particular exponential sum was originally studied by J. Friedlander and H. Iwaniec (1985) with upper bound first obtained by B. Birch and E. Bombieri (1985) by special arguments. Further improvements were made by N. Katz (1986) using l-adic method, and very recently by C. Chen and X. Lin (2022) by p-adic method. In an attempt to understand a remark made by Katz at the end of his 1986 paper, M. Roth and I were able to find a way that leads to a slight improvement on the upper bound of this sum, as well as on a family of similar exponential sums. Our method is based on l-adic cohomology, consisting of finer studying of the local monodromies at zero via representations of the inertia groups there. In this talk, I will survey the above-mentioned results and give a high-level overview of our procedure that gives rise to this new improvement. This is joint work in progress with M. Roth.