We define a Weil-étale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex ) of a large class of -constructible sheaves on an integral1-dimensional proper arithmetic scheme flat over Spec(). 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 -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 -function to the dual of a -constructible sheaf; up to a finite number of factors, this -function is an Artin -function at . Our main theorem contains a vanishing order formula at for the -function and states that, in the tamely ramified case, the special value at is given up to sign by the Euler characteristic. This generalizes the analytic class number formula for the special value at of the Dedekind zeta function. In the function field case, this is a theorem of Geisser–Suzuki.
Cite this article
Adrien Morin, Tori over number fields and special values at . Doc. Math. 28 (2023), no. 1, pp. 173–260DOI 10.4171/DM/906