Joint IAS/Princeton University Number Theory Seminar

Recently, Fargues and Scholze attached a semi-simple L-parameter to any smooth irreducible representation of a p-adic reductive group, realizing the local Langlands correspondence as a geometric Langlands correspondence over the Fargues-Fontaine curve. They conjectured that there should exist an analogue of the geometric Langlands conjecture in this setting, known as the categorical local Langlands correspondence. Concretely, this conjecture translates to the belief that certain Shtuka spaces, generalizing the Lubin-Tate and Drinfeld towers appearing in the work of Harris-Taylor, should have cohomology dictated by the semi-simple L-parameter that they construct. In this talk, we will explain how one can make some progress on this conjecture by showing the Fargues-Scholze correspondence is compatible with known instances of the local Langlands correspondence through global methods, and then using this compatibility together with techniques from geometric Langlands to fully describe the cohomology of these Shtuka spaces in certain cases.