Ana Caraiani — The geometrization of the local Langlands correspondence , after Fargues and Scholze

The Langlands program is an intricate network of conjectures that connect different areas of mathematics, such as number theory, algebraic geometry, and representation theory. There are several flavours of the Langlands program: local and global, arithmetic and geometric. Traditionally, the arithmetic Langlands program, in the setting of p-adic fields and number fields, has not been able to benefit from the flexibility available in other, more geometric settings.

In this talk, we give an overview of the recent work of Fargues and Scholze, which gives a geometrization of the local Langlands correspondence and which works in particular for p-adic fields. One concrete outcome of this work is a general construction of semi-simple local Langlands parameters attached to irreducible smooth representations of p-adic groups.

In fact, the work of Fargues and Scholze gives much more than this construction, by introducing powerful new techniques and structures from the geometric Langlands program to this setting. The key geometric object underlying this work is the moduli stack of vector bundles (or, more generally, G-bundles) on the Fargues–Fontaine curve. We will describe the geometry of this space, its connection to the representation theory of p-adic groups, and give a flavour of the additional structures it allows us to access.

To give a sense of the tremendous impact that the work of Fargues and Scholze has already had on the field, we will end by mentioning a few striking applications that have been developed by various researchers since then: to the representation theory of p-adic groups and to the cohomology of local and global Shimura varieties.