Exponential sums and equidistribution

Organized by Matthew Hase-Liu and Caleb Ji

This is a learning seminar broadly about the sheaf-theoretic approach to equidistribution results for families of exponential sums over finite fields. For families parametrized by a variety, this is closely related to understanding the geometric monodromy groups of the corresponding sheaves. Katz famously proved an average version of the Sato-Tate law for Kloosterman sums in several variables and developed a framework to study exponential sums/arithmetic Fourier transforms of trace functions of perverse sheaves on $\mathbb{G}_a$. Katz and later Forey, Fresan, and Kowalski extended these ideas to families parametrized by multiplicative characters of the points of any connected commutative algebraic group by exploiting the formalism of Tannakian categories and Sawin's quantitative sheaf theory.

We will roughly follow parts of "Arithmetic Fourier transforms over finite fields" by Forey, Fresan, and Kowalski, as well as "Gauss Sums, Kloosterman Sums, and Monodromy Groups" by Katz. Ideally talks will be relatively self-contained, aside from the first few establishing some technical background.

References

Here is a list of the references used in the seminar:

Schedule

We meet on Thursdays from 5:30 to 7:00 in Room 528.

Date Speaker Abstract References
01/30 Matthew Hase-Liu Overview of seminar and introduction to equidistribution theorems: I'll give a motivated introduction to the definition of equidistribution and then discuss some examples related to exponential sums. These ideas go hand in hand with some of the greatest developments in algebraic geometry, and I will try to explain a small part of this story. Notes
02/06 Caleb Ji Deligne's equidistribution theorem: Deligne's equidistribution theorem states that, under certain natural conditions, the Frobenius elements of Galois representations over function fields are equidistributed in a maximal compact subgroup of the associated monodromy group. In this talk we will prove this theorem and give some applications. Notes
02/13 Kevin Chang Gauss sums, Kloosterman sums, and Kloosterman sheaves: We will show how to geometrize exponential sums using the Grothendieck trace formula and the l-adic Fourier transform. Notes
02/20 Vidhu Adhihetty Convolution: We will introduce the general concept of convolution of sheaves on G_m, and state the main theorem regarding this operation. We will aim to give proofs of some parts of the theorem, and then use convolution to define Kloosterman sheaves and explore some of their properties. Notes
02/27 Fan Zhou Tannakian categories: We discuss some basics of Tannakian categories following Deligne and Milne's article.
03/06 Sofia Wood Equidistribution of Kloosterman angles: We introduce Kloosterman angles and state and reformulate an equidistribution theorem for them.
03/13 Caleb Ji Global monodromy of Kloosterman sheaves: Deligne's equidistribution applied to Kloosterman sheaves gives equidistribution of Kloosterman angles once the monodromy groups of the Kloosterman sheaves is computed. We review this setup and outline the computation of these monodromy groups. This turns out to be a lengthy Lie theoretic computation in which G_2 makes an unexpected appearance.
03/20 Matthew Katz's work on Evans and Rudnick sums:
03/27 Kevin Rigid local systems:
05/02 Amadou Quantitative sheaf theory and applications to equidistribution: