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Abstract: It is a fundamental problem in the Langlands program to express the zeta function of a Shimura variety as a product of automorphic L-functions. The work in this area originates with Eichler, Shimura, and others, who studied the Shimura varieties attached to GL2 and its inner forms. Many authors have contributed to the problem since, with the modern conjectural form due to Langlands, Kottwitz, and Rapoport.
In this talk we demonstrate Scholze's adaptation of the so-called Langlands-Kottwitz approach to studying the cohomology of Shimura varieties. Scholze first developed this technique in the case of the modular curve, and he later generalized those methods to compute the semi-simple zeta function of certain ``simple'' Shimura varieties. Our focus will be on this second case. In particular, we will explain the way in which Scholze is able to exploit the particularly nice geometry of the integral models of these Shimura varieties to adapt the approach to cases of bad reduction.