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Motivic Mellin transforms and exponential sums through non-archimedean geometry

Final Report Summary - MOTMELSUM (Motivic Mellin transforms and exponential sums through non-archimedean geometry)

Igusa's conjecture on exponential sums from 1978 predicted upper bounds for certain finite oscillatory sums with index set running over integers modulo powers of primes, called p-adic exponential sums. This conjecture had been widely open ever since its formulation in the introduction of Igusa's book of 1978. Due to this ERC Consolidator grant MOTMELSUM, in work of the PI (Cluckers) with team member Kien Huu Nguyen and external co-author Mircea Mustata, the conjecture is now solved in all cases but the rational singularity case, and even so for a generalization of the conjecture formulated by the PI (Cluckers) and Veys. Meanwhile, on the second theme of the ERC grant related to motivic integration, Casselman, Cely and Hales, and also the PI (Cluckers), Gordon and Halupczok have found striking applications of motivic integration in the Langlands program, by new transfer principles for motivic integrals. Motivic integration is an abstract form of integration closely related to geometry and also to complex, real and p-adic integration. In this ERC project, one of the main themes was motivic integration of oscillatory functions, like Fourier transforms and Mellin transforms. This theme full of challenges has led to a broad development of non-archimedean geometry in various forms. As extra and unexpected outcomes of this study of non-archimedean geometry, the PI has obtained a theory of motivic distributions and wave front sets together with Halupczok, Loeser and Raibaut. The study of non-archimedean geometry has also led to unforeseen applications in number theory, with natural bounds for the number of rational points of bounded height in p-adic fields by the PI with Comte, Forey, and Loeser, similar to the results of Pila and Wilkie over the reals.