Pol van Hoften (VU Amsterdam): Igusa stacks and the cohomology of Shimura varieties

Associated to a modular form f is a two-dimensional Galois representation whose Frobenius eigenvalues can be expressed in terms of the Fourier coefficients of f, using a formula known as the Eichler–Shimura congruence relation. This relation was proved by Eichler–Shimura and Deligne by analyzing the mod p (bad) reduction of the modular curve of level ?0(p). In this talk, I will discuss joint work with Patrick Daniels, Dongryul Kim and Mingjia Zhang, where we give a new proof of this congruence relation that happens “entirely on the generic fibre”. More precisely, we prove a compatibility result between the cohomology of Shimura varieties of Hodge type and the Fargues?Scholze semisimple local Langlands correspondence, generalizing the Eichler–Shimura relation of Blasius–Rogawski. Our proof makes crucial use of the Igusa stacks that we construct, generalizing earlier work of Zhang in the PEL case.

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