Constructible tori over Dedekind schemes

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 $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathbb{Z})$ 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.