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Constructible tori over Dedekind schemes

Adrien Morin, Takashi Suzuki

TL;DR

The paper develops a duality between torsion-free constructible tori over a Dedekind scheme $X$ and torsion-free $\mathbb{Z}$-constructible sheaves, realized through two-term complexes and Cartier duality. It then systematically builds a derived framework of constructible tori, including a robust notion of duals, biduality, and an abelian category of constructible tori with torsion, establishing a functorial determinant theory via det Lie and its extensions. Using these foundations, the authors define $L$-functions for constructible tori via their étale realizations and prove a Weil-étale special value formula at $s=0$, extending prior tame-ramification results. The work unifies local-to-global determinant identities, multiplicativity under exact sequences, Weil restriction behavior, and pushforward properties, providing a comprehensive toolkit for studying arithmetic invariants of tori on Dedekind schemes and their $L$-values.

Abstract

We introduce an exact category of torsion-free constructible tori and an abelian category of constructible tori over a Dedekind scheme with perfect residue fields. The first one has an explicit description as $2$-term complexes of smooth commutative group algebraic spaces. Using the second-named author's duality results arXiv:1806.07641, we prove that they are equivalent to the opposite of the categories of torsion-free $\mathbb{Z}$-constructible sheaves and all $\mathbb{Z}$-constructible sheaves, respectively. We then define $L$-functions for constructible tori over a Dedekind scheme proper over $\mathrm{Spec}(\mathbb{Z})$ in terms of their étale realizations and prove a special value formula at $s=0$ using the Weil-étale formalism developed by the first-named author in arXiv:2210.09102. This extends the results of the first-named author by removing the tame ramification hypothesis.

Constructible tori over Dedekind schemes

TL;DR

The paper develops a duality between torsion-free constructible tori over a Dedekind scheme and torsion-free -constructible sheaves, realized through two-term complexes and Cartier duality. It then systematically builds a derived framework of constructible tori, including a robust notion of duals, biduality, and an abelian category of constructible tori with torsion, establishing a functorial determinant theory via det Lie and its extensions. Using these foundations, the authors define -functions for constructible tori via their étale realizations and prove a Weil-étale special value formula at , extending prior tame-ramification results. The work unifies local-to-global determinant identities, multiplicativity under exact sequences, Weil restriction behavior, and pushforward properties, providing a comprehensive toolkit for studying arithmetic invariants of tori on Dedekind schemes and their -values.

Abstract

We introduce an exact category of torsion-free constructible tori and an abelian category of constructible tori over a Dedekind scheme with perfect residue fields. The first one has an explicit description as -term complexes of smooth commutative group algebraic spaces. Using the second-named author's duality results arXiv:1806.07641, we prove that they are equivalent to the opposite of the categories of torsion-free -constructible sheaves and all -constructible sheaves, respectively. We then define -functions for constructible tori over a Dedekind scheme proper over in terms of their étale realizations and prove a special value formula at using the Weil-étale formalism developed by the first-named author in arXiv:2210.09102. This extends the results of the first-named author by removing the tame ramification hypothesis.
Paper Structure (13 sections, 39 theorems, 98 equations)

This paper contains 13 sections, 39 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Torsion-free constructible tori
  4. Duals of torsion-free Z-constructible sheaves
  5. Comparison with derived Hom to Gm
  6. Duals of torsion-free constructible tori
  7. Biduality
  8. Constructible tori (with torsion)
  9. Multiplicativity of the determinant of the Lie algebra for the Néron model of a torus
  10. The functor detLie for constructible tori
  11. L-functions for constructible tori
  12. The fundamental line and special values at $s=0$
  13. $K$-theory of $\mathbb{Z}$-constructible sheaves

Key Result

Theorem 1

Let $\mathbb{Z} - \mathrm{Con}_{\mathrm{tf}} / X$ denote the category of torsion-free $\mathbb{Z}$-constructible sheaves and $\mathrm{CT}_{\mathrm{tf}} / X$ the category of torsion-free constructible tori. The functors \begin{tikzcd}[ampersand replacement=\&] {\mathbb{Z} \mhyphen \mathrm{Con}_{\ma

Theorems & Definitions (97)

  • Theorem 1: \ref{['0041']}
  • Definition
  • Theorem 2: \ref{['0043']}
  • Definition
  • Proposition : \ref{['prop:extension_detLie']}
  • Theorem 3: \ref{['thm:special_value']}
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • ...and 87 more