Number theory and dynamical systems on foliated spaces. (English) Zbl 1003.11029

Deninger, Christopher

Jahresber. Dtsch. Math.-Ver. 103, No. 3, 79-100 (2001).

This is a report on some ongoing developments in the search for a new understanding of number theoretical zeta functions in terms of dynamical systems and foliations, resp. their cohomology. Central to this theory is a conjectural Lefschetz formula connecting closed orbits of a dynamical system to its action on a foliation cohomology.
The paper deals with three topics. Firstly, it reports on progress on the dynamical Lefschetz formula for foliations of codimension one which is mainly due to Álvarez López and Kordyukov. Next the explicit formulas of number theory are compared with conjectural Lefschetz formulas. Motivated by this comparison, the author finishes with an extension of the conjectural Lefschetz formulas to the case of laminated spaces or solenoids, i.e., spaces which locally look like \(\mathbb{R}^n\) times a totally disconnected space. The author considers solenoids which are foliated by solenoids of lower dimension and formulates a Lefschetz formula in this setting. He gives examples where the formula can be proved.

Reviewer: Anton Deitmar (Exeter)

MSC:

11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11M36	Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
37C27	Periodic orbits of vector fields and flows

Keywords:

zeta functions; dynamical systems; foliations; cohomology; conjectural Lefschetz formula; dynamical Lefschetz formula; foliations of codimension one; solenoids