WEC&DS · Weil-étale cohomology and Deninger's dynamical system

Weil-étale cohomology and Deninger's dynamical system

The first objective of Morin's research proposal was to study the connections between Lichtenbaum's Weil-étale cohomology and Deninger's dynamical system. A second objective was to use this analogy to obtain new results in Weil-étale cohomology. Recall that Weil-étale cohomology (respectively Deninger's programme) is meant to provide an arithmetic cohomology (respectively a geometric cohomology) relevant for the study of motivic L-functions. According to Deninger's programme, a foliated dynamical system should be attached to an arithmetic scheme. This dynamical system would produce Deninger's conjectural cohomological formalism. On the other hand, Lichtenbaum predicts the existence of a Weil-étale cohomology theory for arithmetic schemes allowing a cohomological interpretation for the special values of the corresponding zeta functions. This conjectural Weil-étale cohomology should be the cohomology of a deeper topological structure, namely the conjectural Weil-étale topos. The Weil-étale topos is naturally defined in characteristic p while an unsatisfactory definition has been given for number rings and more generally for arithmetic schemes. To partially achieve the first project objective, we use topos theory in order to study simultaneously the Weil-étale topos and Deninger's dynamical system. We express some basic properties of Deninger's conjectural dynamical system in terms of topoi and morphisms of topoi. Then, we show that the current definition of the Weil-étale topos of a number ring satisfies these properties. In particular, the flow, the closed orbits, the fixed points of the flow and the foliation in positive characteristic are well defined on the Weil-étale topos. This analogy extends to arithmetic schemes. Over a prime number and over the archimedean place of the field of rational numbers, we define a morphism from a topos associated to Deninger's dynamical system to the Weil-étale topos. This morphism is compatible with the structure mentioned above. To achieve the second project objective, we introduce a method which yields a conditional definition for the Weil-étale cohomology. More precisely, Lichtenbaum conjectured the existence of the Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme at s = 0 in terms of Euler-Poincaré characteristics. Assuming the (conjectured) finite generation of some motivic cohomology groups one defines such a cohomology theory for regular schemes proper over the ring of integers. In particular, we compute (unconditionally) the right Weil-étale cohomology of number rings and projective spaces over number rings. This yields a precise version of Lichtenbaum's conjecture, which expresses the vanishing order (resp. the special value up to sign) of the Zeta function of the scheme at s = 0 as the rank (resp. the determinant) of the perfect complex of abelian groups given by Weil-étale cohomology with compact support. This conjecture is shown to be compatible to Soulé's conjecture and to the Tamagawa Number Conjecture. Lichtenbaum's conjecture for projective spaces over the ring of integers of an abelian number field follows. The first objective of Morin's research proposal was to study the connections between Lichtenbaum's Weil-étale cohomology and Deninger's dynamical system. A second objective was to use this analogy to obtain new results in Weil-étale cohomology. Recall that Weil-étale cohomology (respectively Deninger's programme) is meant to provide an arithmetic cohomology (respectively a geometric cohomology) relevant for the study of motivic L-functions. According to Deninger's program, a foliated dynamical system should be attached to an arithmetic scheme. This dynamical system would produce Deninger's conjectural cohomological formalism. On the other hand, Lichtenbaum predicts the existence of a Weil-étale cohomology theory for arithmetic schemes allowing a cohomological interpretation for the special values of the corresponding zeta functions. This conjectural Weil-étale cohomology should be the cohomology of a deeper topological structure, namely the conjectural Weil-étale topos. The Weil-étale topos is naturally defined in characteristic p while an unsatisfactory definition has been given for number rings and more generally for arithmetic schemes. To partially achieve the project's first objective, we use topos theory in order to study simultaneously the Weil-étale topos and Deninger's dynamical system. We express some basic properties of Deninger's conjectural dynamical system in terms of topoi and morphisms of topoi. Then we show that the current definition of the Weil-étale topos of a number ring satisfies these properties. In particular, the flow, the closed orbits, the fixed points of the flow and the foliation in positive characteristic are well defined on the Weil-étale topos. This analogy extends to arithmetic schemes. Over a prime number and over the archimedean place of the field of rational numbers, we define a morphism from a topos associated to Deninger's dynamical system to the Weil-étale topos. This morphism is compatible with the structure mentioned above. To achieve the second project objective, we introduce a method which yields a conditional definition for the Weil-étale cohomology. More precisely, Lichtenbaum conjectured the existence of the Weil-étale cohomology to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme at s = 0 in terms of Euler-Poincaré characteristics. Assuming the (conjectured) finite generation of some motivic cohomology groups one defines such a cohomology theory for regular schemes proper over the ring of integers. In particular, we compute (unconditionally) the right Weil-étale cohomology of number rings and projective spaces over number rings. This yields a precise version of Lichtenbaum's conjecture, which expresses the vanishing order (resp. the special value up to sign) of the Zeta function of the scheme at s = 0 as the rank (resp. the determinant) of the perfect complex of abelian groups given by Weil-étale cohomology with compact support. This conjecture is shown to be compatible to Soulé's conjecture and to the Tamagawa Number Conjecture. Lichtenbaum's conjecture for projective spaces over the ring of integers of an abelian number field follows.

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