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- Centre International de Rencontres Mathématiques
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The thematic month “Arithmetic and Information Theory” focuses on arithmetic geometry, information theory and their interplay.
Arithmetic geometry is a fundamental and growing area of research in modern mathematics, deeply connected to almost all of its branches. As a measure of its importance, about a quarter of the Fields medalists have worked on problems in arithmetic geometry. There is a vast number of different techniques that have been developed in the field. Algebraic Geometry Codes Theory studies (amongst others) varieties, and in particular curves, with many rational points, having in mind that maximal curves are the aspirational target to obtain codes with the best correction rate. This maximality corresponds to extremal points in the distribution of the Frobenius endomorphism acting on the Tate module of the Jacobian of the curves. For elliptic curves, we are confronted with the Sato-Tate conjecture which is concerned with the statistics to modulo p reductions of families of elliptic curves defined over the field of rational numbers. Galois representations on ´etale cohomology play a central role in this field. The most remarkable aspects of this series of conferences is the omnipresent exchange between applied (codes, cryptography) and pure (arithmetic and geometry) mathematics. The first branch supplies the second with new problems. For example, the impulse of Serre, Manin, and Ihara, motivated by the introduction of Goppa codes, led to hundreds of articles devoted to the study of the number of points on curves over finite fields. A more specialized and recent example concerns quantum codes that lead to the study of couples code/sub-code having special properties. As a counterpart, arithmetic and geometry allow remarkable new results to be obtained in their applications. For example, cryptosystems constructed from elliptic curves over finite fields, or the best asymptotically excellent codes constructed using algebraic varieties.
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