Tori over number fields and special values at $s=1$

Abstract

We define a Weil-étale 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 integral1-dimensional proper arithmetic scheme flat over Spec($\mathbb{Z}$). This complex can be thought of as computing Weil-étale homology. For those 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-étale 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 is a theorem of Geisser–Suzuki.