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Shimura varieties and gerbes

Robert P. Langlands, Michael Rapoport

TL;DR

Langlands and Rapoport formulate a conjectural description of the mod $p$ reduction of Shimura varieties by constructing a pseudomotivic Galois group and a motivic Galois group via gerbes and Tannakian categories. They develop explicit gerbes attached to tori and CM motives, establish universal properties and functorialities, and relate them to endoscopic stabilization and Kottwitz invariants. The core contribution is a group-theoretic, conjectural formula for the count of points on the reduced Shimura variety in terms of stable conjugacy data and orbital integrals, together with a framework connecting motives, Frobenius actions, and Shimura data under standard conjectures. The work integrates Galois cohomology, Tate–Nakayama theory, and Tannakian formalism to bridge arithmetic geometry, automorphic forms, and motivic philosophy, laying groundwork for future proofs and refinements (notably by Kottwitz, Kisin, and others).

Abstract

This is the English translation, done by Yihang Zhu, of the paper by Langlands and Rapoport, originally published in J. Reine Angew. Math. 378 (1987), pages 113-220. The translator also added some historical notes.

Shimura varieties and gerbes

TL;DR

Langlands and Rapoport formulate a conjectural description of the mod reduction of Shimura varieties by constructing a pseudomotivic Galois group and a motivic Galois group via gerbes and Tannakian categories. They develop explicit gerbes attached to tori and CM motives, establish universal properties and functorialities, and relate them to endoscopic stabilization and Kottwitz invariants. The core contribution is a group-theoretic, conjectural formula for the count of points on the reduced Shimura variety in terms of stable conjugacy data and orbital integrals, together with a framework connecting motives, Frobenius actions, and Shimura data under standard conjectures. The work integrates Galois cohomology, Tate–Nakayama theory, and Tannakian formalism to bridge arithmetic geometry, automorphic forms, and motivic philosophy, laying groundwork for future proofs and refinements (notably by Kottwitz, Kisin, and others).

Abstract

This is the English translation, done by Yihang Zhu, of the paper by Langlands and Rapoport, originally published in J. Reine Angew. Math. 378 (1987), pages 113-220. The translator also added some historical notes.

Paper Structure

This paper contains 7 sections, 52 theorems, 552 equations.

Table of Contents

  1. Introduction
  2. Cohomological constructions
  3. The pseudomotivic Galois group
  4. The motivic Galois group
  5. A conjecture on Shimura varieties and its consequences
  6. The conjecture as a consequence of the Standard Conjectures in some cases
  7. Two examples

Key Result

Lemma 2.1

Let $\phi: \mathscr{D}^K \to \mathscr{G}$ be a homomorphism of gerbes. Then there exists an unramified extension $L$ of $\mathbb{Q}_p$, a homomorphism $\psi: \mathscr{D}^L \to \mathscr{G}$, and an extension $K_1$ containing both $K$ and $L$, such that the following diagram commutes: \xymatrix{ \m

Theorems & Definitions (58)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 48 more