Shimura Varieties and Eigensheaves

The cohomology of Shimura varieties is a fundamental object of study in algebraic number theory by virtue of the fact that it is the only known geometric realization of the global Langlands correspondence over number fields. Usually, the cohomology is computed through very delicate techniques involving the trace formula. However, this perspective has several limitations, especially with regards to questions concerning torsion. In this talk, we will discuss a new paradigm for computing the cohomology of Shimura varieties by decomposing certain sheaves coming from Igusa varieties into Hecke eigensheaves on the moduli stack of G-bundles on the Fargues-Fontaine curve. Using this point of view, we will describe several conjectures on the torsion cohomology of Shimura varieties after localizing at suitably “generic” L-parameters, as well as some known results in the case that the parameter factors through a maximal torus. Motivated by this, we will sketch part of an emerging picture for describing the cohomology beyond this generic locus by considering certain “generalized eigensheaves” whose eigenvalues are spread out in multiple cohomological degrees based on the size of a certain Arthur SL_{2} in a way that is reminiscent of Arthur’s cohomological conjectures on the intersection cohomology of Shimura Varieties. This is based on joint work with Lee, joint work in progress with Caraiani and Zhang, and conversations with Bertoloni-Meli and Koshikawa.