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arXiv · 2409.07363

Geometric Eisenstein series I: finiteness theorems

Linus Hamann·David Hansen·Peter Scholze
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Abstract

We develop the theory of geometric Eisenstein series and constant term functors for $\ell$-adic sheaves on stacks of bundles on the Fargues-Fontaine curve. In particular, we prove essentially optimal finiteness theorems for these functors, analogous to the usual finiteness properties of parabolic inductions and Jacquet modules. We also prove a geometric form of Bernstein's second adjointness theorem, generalizing the classical result and its recent extension to more general coefficient rings proved in [Dat-Helm-Kurinczuk-Moss]. As applications, we decompose the category of sheaves on $\mathrm{Bun}_G$ into cuspidal and Eisenstein parts, and show that the gluing functors between strata of $\mathrm{Bun}_G$ are continuous in a very strong sense.

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Linus Hamann, David Hansen, Peter Scholze. 2024-09-11. Geometric Eisenstein series I: finiteness theorems. https://arxiv.org/abs/2409.07363

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