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

Adrien Morin

Documenta mathematica, Tome 28 (2023) no. 1, pp. 173-260

Résumé

We define a Weil-étale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex Zc) of a large class of Z-constructible sheaves on an integral1-dimensional proper arithmetic scheme flat over Spec(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 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 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.

Détail

DOI : 10.4171/dm/906

Classification : 14G10, 11G40, 11R42, 14F20, 14F42
Mots-clés : Weil-étale cohomology, special values, L-functions