Tori over number fields and special values at s=1

Abstract

We define a Weil-\'etale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex $\mathbb{Z}^c$) of a large class of $\mathbb{Z}$-constructible sheaves on an integral $1$-dimensional proper arithmetic scheme flat over $\mathrm{Spec}(\mathbb{Z})$. This complex can be thought of as computing Weil-\'etale homology. For those $\mathbb{Z}$-constructible sheaves that are moreover tamely ramified, we define an "additive" complex which we think of as the Lie algebra of the dual of the $\mathbb{Z}$-constructible sheaf. The product of the determinants of the additive and Weil-\'etale complex is called the fundamental line. We prove a duality theorem which implies that the fundamental line has a natural trivialization, giving a multiplicative Euler characteristic. We attach a natural $L$-function to the dual of a $\mathbb{Z}$-constructible sheaf; up to a finite number of factors, this $L$-function is an Artin $L$-function at $s+1$. Our main theorem contains a vanishing order formula at $s=0$ for the $L$-function and states that, in the tamely ramified case, the special value at $s=0$ is given up to sign by the Euler characteristic. This generalizes the analytic class number formula for the special value at $s=1$ of the Dedekind zeta function. In the function field case, this a theorem of arXiv:2009.14504.