Geometric Eisenstein series I: finiteness theorems

TL;DR

This paper develops the theory of geometric Eisenstein series on the Fargues-Fontaine curve, proving finiteness theorems and a geometric Bernstein's second adjointness, with applications to the structure of sheaves on moduli stacks.

Contribution

It introduces finiteness theorems and a geometric Bernstein's second adjointness for geometric Eisenstein series on stacks of bundles, extending classical results to a geometric setting.

Findings

Proved finiteness theorems for functors related to geometric Eisenstein series.

Established a geometric form of Bernstein's second adjointness theorem.

Decomposed sheaf categories on Bun_G into cuspidal and Eisenstein parts, showing strong continuity of gluing functors.

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.