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Drinfeld's lemma for $F$-isocrystals, I

Kiran S. Kedlaya

Abstract

We prove that in either the convergent or overconvergent setting, an absolutely irreducible $F$-isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic $p$, further equipped with actions of the partial Frobenius maps, is an external product of $F$-isocrystals over the multiplicands. The corresponding statement for lisse $\bar{\mathbb{Q}}_\ell$-sheaves, for $\ell \neq p$ a prime, is a consequence of Drinfeld's lemma on the fundamental groups of absolute products of schemes in characteristic $p$. The latter plays a key role in V. Lafforgue's approach to the Langlands correspondence for reductive groups with $\ell$-adic coefficients; the $p$-adic analogue will be considered in subsequent work with Daxin Xu.

Drinfeld's lemma for $F$-isocrystals, I

Abstract

We prove that in either the convergent or overconvergent setting, an absolutely irreducible -isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic , further equipped with actions of the partial Frobenius maps, is an external product of -isocrystals over the multiplicands. The corresponding statement for lisse -sheaves, for a prime, is a consequence of Drinfeld's lemma on the fundamental groups of absolute products of schemes in characteristic . The latter plays a key role in V. Lafforgue's approach to the Langlands correspondence for reductive groups with -adic coefficients; the -adic analogue will be considered in subsequent work with Daxin Xu.
Paper Structure (11 sections, 42 theorems, 28 equations)

This paper contains 11 sections, 42 theorems, 28 equations.

Table of Contents

  1. Local Drinfeld's lemma for schemes
  2. Products over $\mathbb{F}_p$
  3. Drinfeld's lemma for schemes
  4. Convergent $\Phi$-isocrystals
  5. External products
  6. Stratification by the diagonal Newton polygon
  7. Relative Dieudonné--Manin
  8. Joint Newton slopes
  9. Partial slope filtrations
  10. Slices
  11. Overconvergent $\Phi$-isocrystals

Key Result

Lemma 1.2

The base extension functor $\mathop{\mathrm{\mathbf{FEt}}}\nolimits(X) \to \mathop{\mathrm{\mathbf{FEt}}}\nolimits(X_k/\varphi_k)$ is an equivalence of categories, with quasi-inverse given by taking $\varphi_k$-invariants.

Theorems & Definitions (108)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 98 more