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Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties

Mark Kisin, Rong Zhou

Abstract

Let $A$ be an abelian variety over a number field $\mathrm E\subset \mathbb C$ and let $\mathbf G$ denote the Mumford--Tate group of $A$. After replacing $\mathrm E$ by a finite extension, the action of the absolute Galois group $\mathrm{Gal}(\overline{\mathrm E}/\mathrm E)$ on the $\ell$-adic cohomology $\mathrm{H}^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ factors through $\mathbf G(\mathbb Q_\ell).$ We show that for $v$ an odd prime of $\mathrm E$ where $A$ has good reduction, the conjugacy class of Frobenius $\mathrm{Frob}_v$ in $\mathbf G(\mathbb Q_\ell)$ is independent of $\ell$. Along the way we prove that every point in the $μ$-ordinary locus of the special fiber of Shimura varieties has a special point lifting it.

Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties

Abstract

Let be an abelian variety over a number field and let denote the Mumford--Tate group of . After replacing by a finite extension, the action of the absolute Galois group on the -adic cohomology factors through We show that for an odd prime of where has good reduction, the conjugacy class of Frobenius in is independent of . Along the way we prove that every point in the -ordinary locus of the special fiber of Shimura varieties has a special point lifting it.

Paper Structure

This paper contains 25 sections, 52 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Group theoretic results
  3. $\sigma$-straight elements
  4. $\mu$-ordinary $\sigma$-conjugacy classes
  5. Parahoric group schemes
  6. Néron models of tori
  7. Deformation theory of $p$-divisible groups
  8. Local models and good embeddings
  9. The versal deformation space with tensors
  10. Deformations with étale tensors
  11. Canonical liftings for $\mu$-ordinary $p$-divisible groups
  12. Integral models of Shimura varieties and canonical liftings
  13. Integral models
  14. Strongly acceptable Shimura varieties
  15. Some functorial properties of integral models
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $p>2$ and $v|p$ a prime of $\mathrm{E}$ where $A$ has good reduction. Then there exists $\gamma\in \mathrm{Conj}_{\mathbf{G}}(\mathbb{Q})$ such that

Theorems & Definitions (122)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1.7
  • proof
  • Lemma 2.1.9
  • proof
  • Definition 2.2.4
  • Remark 2.2.5
  • Lemma 2.2.6
  • ...and 112 more