Lectures on Applied $\ell$-adic Cohomology

TL;DR

This paper discusses how advanced $ ext{ extlbrackdbl} ext{ extrbrackdbl}$-adic cohomology techniques, developed by Grothendieck, Deligne, Katz, and Laumon, can address classical problems in analytic number theory.

Contribution

It systematically applies deep $ ext{ extlbrackdbl} ext{ extrbrackdbl}$-adic cohomology methods to make progress on longstanding questions in analytic number theory.

Findings

Demonstrates new applications of $ ext{ extlbrackdbl} ext{ extrbrackdbl}$-adic cohomology in number theory

Provides insights into classical problems using modern cohomological techniques

Extends previous theoretical frameworks in $ ext{ extlbrackdbl} ext{ extrbrackdbl}$-adic cohomology

Abstract

We describe how a systematic use the deep methods from $\ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 Arizona Winter School.