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This lecture by Sophie Morel (ENS Lyon) explores the construction of Galois representations following Scholze's approach. Dive into the fascinating connection between the cohomology of hyperbolic 3-dimensional varieties and Galois theory. Learn how the Langlands program predicts that the singular cohomology of a hyperbolic variety X (a quotient of hyperbolic space by an "arithmetic" group of isometries) with coefficients in Z/nZ naturally carries an action of the absolute Galois group of a number field—a surprising prediction since X is not an algebraic variety. Discover the key insight of connecting the torsion cohomology of X to that of another locally symmetric space which happens to be a Shimura variety, and therefore an algebraic variety defined over a number field. The presentation examines how this idea was independently implemented by Harris-Lan-Taylor-Thorne, Scholze, and Boxer (in chronological order), with a specific focus on Scholze's approach to this problem that extends beyond the specific case presented.
