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

Basic data for this project

Description

The aim of our research proposal is to study the connections between Lichtenbaum's Weil-etale cohomology and Deninger's dynamical system. Weil-etale cohomology (respectively Deninger's program) is meant to provide an arithmetic cohomology (respectively a geometric cohomology) relevant for the study of motivic L-functions. These are two very ambitious and promising directions in arithmetic geometry, which are certainly strongly related even if these connections are not well understood. 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-etale cohomology theory for arithmetic schemes allowing a cohomological interpretation for the special values of the corresponding zeta functions. This conjectural Weil-etale cohomology should be the cohomology of a deeper topological structure, namely the conjectural Weil-etale topos. The Weil-etale topos is naturally defined in characteristic p while an unsatisfactory definition has been given for number rings and more generally for arithmetic schemes. This Weil-etale topos, i.e. this generalized space, turns out to be closely related to Deninger's dynamical system. We propose to use topos theory in order to study simultaneously the Weil-etale topos and Deninger's dynamical system. The insight provided by Deninger's work will be applied to obtain new results in Weil-etale cohomology. Respectively, Lichtenbaum's explicit computations will be used to obtain information on Deninger's dynamical system. The ultimate goal of this research project is to define and study the conjectural Weil-etale topos in characteristic zero. One interesting aspect of this project is the interaction of general topos theory and dynamical systems with more classical and well etablished number theory, such as the analytic class number formula.