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The stable trace formula for Igusa varieties, II

Alexander Bertoloni Meli, Sug Woo Shin

Abstract

Assuming the trace formula for Igusa varieties in characteristic p, which is known by Mack-Crane in the case of Hodge type with good reduction at p, we stabilize the formula via Kaletha's theory of rigid inner twists when the reductive group in the underlying Shimura datum is quasi-split at p. This generalizes our earlier work under more restrictive hypotheses.

The stable trace formula for Igusa varieties, II

Abstract

Assuming the trace formula for Igusa varieties in characteristic p, which is known by Mack-Crane in the case of Hodge type with good reduction at p, we stabilize the formula via Kaletha's theory of rigid inner twists when the reductive group in the underlying Shimura datum is quasi-split at p. This generalizes our earlier work under more restrictive hypotheses.
Paper Structure (63 sections, 29 theorems, 150 equations)

This paper contains 63 sections, 29 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Igusa varieties
  3. The main theorem
  4. Sketch of proof
  5. A guide for the reader
  6. Notation and conventions
  7. Preliminaries
  8. Rigid inner twists
  9. The local gerbe
  10. The global gerbe
  11. Localization
  12. Local rigid inner twists
  13. Isocrystals vs local rigid inner twists
  14. Review of the Kottwitz set $B(G)$
  15. Rigid inner twists vs extended pure inner twists
  16. ...and 48 more sections

Key Result

Theorem 1.2.2

Keep the hypotheses in the bullet list of §ss:Igusa-intro and assume Conjecture conj:Igusa-simplified. For $\phi^{\infty,p}$, $\phi_p$, $\xi$, and sufficiently large $j$ as in the conjecture, there exists $h\in {\mathcal{H}}(H(\mathbb{A}))$ (depending on $j$) such that the following stabilized formu where $\iota({\mathfrak e})\in \mathbb{Q}_{>0}$ is an explicit constant.

Theorems & Definitions (93)

  • Conjecture 1.2.1
  • Theorem 1.2.2
  • Proposition 2.1.2: KalethaRigidvsIsoc
  • Lemma 2.1.6
  • proof
  • Definition 2.1.7
  • Remark 2.1.8
  • Lemma 2.1.9
  • proof
  • Lemma 2.1.10
  • ...and 83 more