Geometric Eisenstein series and the Fargues-Fontaine curve
Linus Hamann (Princeton University)
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 $\operatorname{Bun}_{M}$ of $M$-bundles on a smooth projective curve to eigensheaves on the moduli stack $\operatorname{Bun}_{G}$ 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 $S_{\chi}$ on $\operatorname{Bun}_{T}$ attached to a smooth character $\chi$ of the maximal torus $T$, one can construct an eigensheaf on $\operatorname{Bun}_{G}$ under a certain genericity hypothesis on $\chi$, by applying a geometric Eisenstein functor to $S_{\chi}$. 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 $\chi$. 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.
algebraic geometrynumber theory
Audience: researchers in the topic
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Organizers: | Edgar Costa*, Siyan Daniel Li-Huerta*, Bjorn Poonen*, David Roe*, Andrew Sutherland*, Robin Zhang*, Wei Zhang*, Shiva Chidambaram* |
*contact for this listing |