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Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications

Cong Xue

Abstract

In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism.

Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications

Abstract

In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism.

Paper Structure

This paper contains 18 sections, 39 theorems, 177 equations.

Table of Contents

  1. Example of automorphic forms for $SL_2$
  2. Proof of Theorem \ref{['thm-coho-cht-is-Hecke-mod-type-fini-Hecke-en-v']}
  3. Reminder of cohomologies of stacks of shtukas
  4. Strategy of the proof of Theorem \ref{['thm-coho-cht-is-Hecke-mod-type-fini-Hecke-en-v']}
  5. Cohomology of stacks of $M$-shtukas and constant term morphism
  6. Action of Hecke operators
  7. Proof of Lemma \ref{['lem-Hecke-operator-on-filtration-G']}
  8. Global excursion operators
  9. Specialization morphism
  10. Drinfeld's lemma
  11. More on Drinfeld's lemma
  12. Excursion operators
  13. Proof of Lemma \ref{['lem-Frob-partiel-commute-with-gamma-excursion-op']}
  14. Langlands parametrizations
  15. Excursion operators on cohomology groups
  16. ...and 3 more sections

Key Result

Proposition 1

For any place $u$ of $X \smallsetminus N$, the vector space $C_c( G(F) \backslash G(\mathbb A) / K_N \Xi , E)$ is a $\mathscr H_{G, u}$-module of finite type.

Theorems & Definitions (69)

  • Proposition 1
  • Theorem 2
  • Lemma 1.0.4
  • Lemma 1.0.7
  • Lemma 1.0.8
  • Lemma 1.0.13
  • Remark 1.0.14
  • Remark 2.1.4
  • Definition 2.1.8: cf. définition 4.7 in vincent and Definition 2.5.1 in cusp-coho
  • Remark 2.1.9
  • ...and 59 more