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Strongly compatible systems associated to semistable abelian varieties

Mark Kisin, Rong Zhou

TL;DR

The paper develops a motivic refinement of strongly compatible systems for abelian varieties by proving that, after a finite base change, the G-valued ℓ-adic representations attached to $H^1_{ ext{ét}}(A_{ar E},oldsymbol{Q}_ullet)$ form a strongly compatible system with respect to the Mumford–Tate group $oldsymbol{G}$, including semistable places and $v|ullet$ cases. Central to the approach are integral and toroidal models of Shimura varieties of Hodge type, p-adic shtukas, and a function-field reduction via Lafforgue and Abe-type results, which yield cross-characteristic Weil–Deligne compatibility. Key contributions include (i) a robust construction of PR integral models with CM lifts and ℓ-independence along all places, (ii) the development of G-valued WD representations, their comparison across characteristics, and their relation to isocrystals, and (iii) the establishment of ℓ-independence for WD data attached to abelian varieties at semistable and boundary points, with concrete applications to Shimura varieties beyond strong admissibility, including vHoften-Ordinary phenomena and Hecke-orbit conjectures. The results provide a unifying framework linking ℓ-adic, p-adic, and function-field techniques to obtain uniform WD invariants and broadening the scope of known compatibility phenomena in arithmetic geometry.

Abstract

We prove a motivic refinement of a result of Weil, Deligne and Raynaud on the existence of strongly compatible systems associated to abelian varieties. More precisely, given an abelian variety $A$ over a number field $\mathrm{E}\subset \mathbb C$, we prove that after replacing $\mathbb E$ by a finite extension, the action of $\mathrm{Gal}(\overline{\mathrm E}/\mathrm E)$ on the $\ell$-adic cohomology $\mathrm H^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ gives rise to a strongly compatible system of $\ell$-adic representations valued in the Mumford--Tate group $\mathbf G$ of $A$. This involves an independence of $\ell$-statement for the Weil--Deligne representation associated to $A$ at places of semistable reduction, extending previous work of ours at places of good reduction.

Strongly compatible systems associated to semistable abelian varieties

TL;DR

The paper develops a motivic refinement of strongly compatible systems for abelian varieties by proving that, after a finite base change, the G-valued ℓ-adic representations attached to form a strongly compatible system with respect to the Mumford–Tate group , including semistable places and cases. Central to the approach are integral and toroidal models of Shimura varieties of Hodge type, p-adic shtukas, and a function-field reduction via Lafforgue and Abe-type results, which yield cross-characteristic Weil–Deligne compatibility. Key contributions include (i) a robust construction of PR integral models with CM lifts and ℓ-independence along all places, (ii) the development of G-valued WD representations, their comparison across characteristics, and their relation to isocrystals, and (iii) the establishment of ℓ-independence for WD data attached to abelian varieties at semistable and boundary points, with concrete applications to Shimura varieties beyond strong admissibility, including vHoften-Ordinary phenomena and Hecke-orbit conjectures. The results provide a unifying framework linking ℓ-adic, p-adic, and function-field techniques to obtain uniform WD invariants and broadening the scope of known compatibility phenomena in arithmetic geometry.

Abstract

We prove a motivic refinement of a result of Weil, Deligne and Raynaud on the existence of strongly compatible systems associated to abelian varieties. More precisely, given an abelian variety over a number field , we prove that after replacing by a finite extension, the action of on the -adic cohomology gives rise to a strongly compatible system of -adic representations valued in the Mumford--Tate group of . This involves an independence of -statement for the Weil--Deligne representation associated to at places of semistable reduction, extending previous work of ours at places of good reduction.
Paper Structure (18 sections, 36 theorems, 138 equations)

This paper contains 18 sections, 36 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Integral models of Shimura varieties
  3. Pappas--Rapoport integral models
  4. Isogeny classes and CM lifts
  5. Independence of $\ell$
  6. Toroidal compactifications
  7. Integral models of toroidal compactifications
  8. Existence of boundary curves
  9. Weil--Deligne representations
  10. $\mathbf{G}$-valued Weil--Deligne representations
  11. Comparison of tame fundamental groups
  12. Weil--Deligne representations associated to isocrystals
  13. Local global compatibility in Lafforgue's theorem
  14. Independence of $\ell$
  15. Local monodromy at the boundary
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $v$ be a place where $A$ has semistable reduction. There exists a $\mathbf{G}$-valued Weil--Deligne representation $\rho_{A,v}^{\mathrm {WD},\mathbf{G}}$ defined over $\mathbb{Q}$ such that

Theorems & Definitions (83)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 2.1.6
  • Lemma 2.1.8
  • proof
  • Proposition 2.1.9
  • proof
  • Theorem 2.1.12: PRshtukas
  • ...and 73 more