**Title:** Applied l-adic cohomology

**Abstract:** The notion of congruence (modulo an integer $q$) was formalised by C. F. Gauss in his Disquisitiones arithmeticae. This is a basic yet fundamental concept in all aspects of number theory. Indeed congruences allow to evaluate and compare integers in way considerably richer than the archimedean order alone permits.

In analytic number theory, several outstanding question -starting with Dirichlet’s theorem on primes in arithmetic progressions- reduce to the of measuring whether some classical arithmetic function (say the characteristic function of prime numbers) correlate with suitable $q$ periodic functions for instance Gauss sums, Jacobi sums or Kloosterman sums. It turns out that these functions, when the modulus $q$ is a prime (to which one can reduce via the Chinese Reminder Theorem) can be recognised as « trace functions». The study of trace functions was initiated by A. Weil in the 1940’s and was pursued by A. Grothendieck in the second half of the century with his refoundation of algebraic geometry and the invention of étale cohomology; it culminated with P. Deligne’s proof of the Riemann Hypothesis for algebraic varieties over finite fields.

In these lectures, we will explain, through various examples, how the theory of trace functions and its subsequent developments in l-adic cohomology, notably the works of Katz and Laumon, have made their way into modern analytic number theory. Several (but not all) of the examples discussed during these lectures originate from joint works with E. Fouvry, E. Kowalski and W. Sawin.

**Lecture 1:** 29 October 2024, 1600 to 1730

**Lecture 2:** 30 October 2024, 1600 to 1730

**Lecture 3:** 01 November 2024, 1115 to 1245

**Lecture 4:** 01 November 2024, 1600 to 1730

**About the speaker:** Ph. Michel’s main research interests concern analytic number theory and range over a variety of techniques and methods which include: arithmetic geometry, exponential sums, sieve methods, automorphic forms and allied representations, L-functions, and ergodic theory.

Ph. Michel obtained his PhD from Université Paris-Saclay in 1995 under the guidance of E. Fouvry. From 1995 to 1998 he was assistant professor at Université Paris-Saclay, then full professor at Université de Montpellier until 2008 then when he moved to Switzerland to join the Ecole Polytechique Fédérale de Lausanne. Ph. Michel was awarded the Peccot-Vimont prize, has been member of the Institut Universitaire de France and was invited speaker at the 2006 International Congress of Mathematician. He is member of the Academia Europaea (Academy of Europe) since 2011 and Fellow of the American Math. Society since 2012.

This lecture series is part of the program "Circle Method and Related Topics"