Analogies between analysis on foliated spaces and arithmetic geometry

Abstract

We point out analogies between (a) the explicit formulas in analytic number theory and transversal index theory, (b) Lichtenbaum's recent conjectures on special values of Hasse-Weil zeta functions and a formula for special values of Ruelle zeta functions, (c) work of Cramer on the zeroes of the Riemann zeta function and a result of Chazarain on the trace of a wave operator. These analogies suggest certain problems in the analysis on foliated spaces. The paper is mostly a review of previous work but the ideas concerning (c) are new.

to appear in a conference volume in honour of Hermann Weyl

References in corpus (5)

• Number theory and dynamical systems on foliated spaces
• A dynamical systems analogue of Lichtenbaum's conjectures on special values of Hasse-Weil zeta functions
• The trace formula for transversally elliptic operators on Riemannian foliations
• The Weil-Etale Topology for Number Rings
• A remark on a relation between foliations and number theory

Cited by in corpus (3)

• On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds
• On Analytical and Topological Properties of Separatrices in 1-D Holomorphic Dynamical Systems and Complex-Time Newton Flows
• Leafwise cohomological expression of dynamical zeta functions on foliated dynamical systems