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Duality for the condensed Weil-étale realisation of $1$-motives over $p$-adic fields

Marco Artusa

TL;DR

This work constructs a topologically faithful local duality for 1-motives over a $p$-adic field by replacing classical Galois cohomology with condensed Weil-étale cohomology and introducing condensed Weil-étale realisations for group schemes and 1-motives. Central to the approach is the functor $R\mathbb{\Gamma}(B_{\hat{W}_F},-)$ valued in the condensed setting, together with a canonical $\mathbb{R}/\mathbb{Z}$-twist that enables Pontryagin duality to be expressed at the level of cohomology objects in $\mathbf{FLCA}$. The paper proves perfect cup-product pairings for tori, abelian varieties, and general 1-motives, and provides detailed structure theorems for the condensed cohomology groups, including a decomposition into locally compact and non-Hausdorff parts. It also explains how these condensed dualities specialize to, and improve upon, classical Tate duality and Harari–Szamuely dualities, while offering a robust framework that preserves topology throughout. Overall, the results yield a refined, topology-aware local duality for algebro-geometric coefficients in the $p$-adic setting with potential for broader generalisations.

Abstract

We extend Tate duality for Galois cohomology of abelian varieties to $1$-motives over a $p$-adic field, improving a result of Harari and Szamuely. To do this, we replace Galois cohomology with the condensed cohomology of the Weil group. This is a topological cohomology theory defined in a previous work, which keeps track of the topology of the $p$-adic field. To see $1$-motives as coefficients of this cohomology theory, we introduce their condensed Weil-étale realisation. Our duality takes the form of a Pontryagin duality between locally compact abelian groups.

Duality for the condensed Weil-étale realisation of $1$-motives over $p$-adic fields

TL;DR

This work constructs a topologically faithful local duality for 1-motives over a -adic field by replacing classical Galois cohomology with condensed Weil-étale cohomology and introducing condensed Weil-étale realisations for group schemes and 1-motives. Central to the approach is the functor valued in the condensed setting, together with a canonical -twist that enables Pontryagin duality to be expressed at the level of cohomology objects in . The paper proves perfect cup-product pairings for tori, abelian varieties, and general 1-motives, and provides detailed structure theorems for the condensed cohomology groups, including a decomposition into locally compact and non-Hausdorff parts. It also explains how these condensed dualities specialize to, and improve upon, classical Tate duality and Harari–Szamuely dualities, while offering a robust framework that preserves topology throughout. Overall, the results yield a refined, topology-aware local duality for algebro-geometric coefficients in the -adic setting with potential for broader generalisations.

Abstract

We extend Tate duality for Galois cohomology of abelian varieties to -motives over a -adic field, improving a result of Harari and Szamuely. To do this, we replace Galois cohomology with the condensed cohomology of the Weil group. This is a topological cohomology theory defined in a previous work, which keeps track of the topology of the -adic field. To see -motives as coefficients of this cohomology theory, we introduce their condensed Weil-étale realisation. Our duality takes the form of a Pontryagin duality between locally compact abelian groups.

Paper Structure

This paper contains 41 sections, 58 theorems, 436 equations.

Table of Contents

  1. Introduction
  2. Relation with previous work
  3. Outline
  4. Preliminaries and notation
  5. Set-theoretical conventions
  6. Topos theory
  7. inftyderived categories
  8. Locally compact abelian groups and Pontryagin duality
  9. Condensed Mathematics
  10. The Weil group of a p-adic field
  11. The condensed cohomology of the Weil group
  12. Group schemes and Cartier duality
  13. Acknowledgements
  14. Condensed Weil-étale realisation
  15. Realisation of algebraic groups
  16. ...and 26 more sections

Key Result

Theorem A

Let $A$ be an abelian variety over a $p$-adic field $F$ and let $A^*$ be its dual abelian variety. For $q=0,1$, there is a perfect pairing between a profinite abelian group and a torsion abelian group.

Theorems & Definitions (168)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G: Tate local duality
  • Definition 1.1
  • Definition 2.1
  • Example 2.2
  • ...and 158 more