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 $\Gamma_0(p)$. 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 rigid generic fibre". More precisely, we prove a compatibility result between the cohomology of Shimura varieties of abelian 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, ourselves, and Kim.