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Explore the first lecture in a series on applied l-adic cohomology delivered by renowned mathematician Philippe Michel at ICTS Bengaluru. Delve into the fundamental concept of congruence in number theory, first formalized by C.F. Gauss, and discover how it enables richer evaluation and comparison of integers beyond archimedean order. Learn how trace functions, developed by A. Weil in the 1940s and advanced through Grothendieck's étale cohomology, have become instrumental in modern analytic number theory. Examine various examples demonstrating the application of trace functions and l-adic cohomology, including collaborative work with E. Fouvry, E. Kowalski, and W. Sawin. Benefit from the expertise of Michel, an accomplished researcher in analytic number theory, whose work spans arithmetic geometry, exponential sums, sieve methods, automorphic forms, L-functions, and ergodic theory.
