Speaker:  Linus Hamann (Princeton University)

Title:  Geometric Eisenstein series and the Fargues-Fontaine curve

Abstract:  Given a connected reductive group G and a Levi subgroup M, Braverman-Gaitsgory and Laumon constructed geometric Eisenstein functors which take Hecke eigensheaves on the moduli stack Bun_M of M-bundles on a smooth projective curve to eigensheaves on the moduli stack Bun _G of G-bundles. Recently, Fargues and Scholze constructed a general candidate for the local Langlands correspondence by doing geometric Langlands on the Fargues-Fontaine curve. In this talk, we explain recent work on carrying the theory of geometric Eisenstein series over to the Fargues-Scholze setting. In particular, we explain how, given the eigensheaf S_x on Bun_T attached to a smooth character χ of the maximal torus T, one can construct an eigensheaf on Bun_G under a certain genericity hypothesis on χ, by applying a geometric Eisenstein functor to S_x. Assuming the Fargues-Scholze correspondence satisfies certain expected properties, we fully describe the stalks of this eigensheaf in terms of normalized parabolic inductions of the generic χ. This eigensheaf has several useful applications to the study of the cohomology of local and global Shimura varieties, and time permitting we will explain such applications. This is in-person only. Masks are encouraged, but not required.