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For the arithmetic study of varieties over finite fields powerful cohomological methods are available which in particular shed much light on the nature of the corresponding zeta functions. For algebraic schemes over spec ℤ and in particular for the Riemann zeta function no cohomology theory has yet been developed that could serve similar purposes. For a long time it had even been a mystery how the formalism of such a theory could look like. This was clarified in [D1]. However until now the conjectured cohomology has not been constructed.
There is a simple class of dynamical systems on foliated manifolds whose reduced leafwise cohomology has several of the expected structural properties of the desired cohomology for algebraic schemes. In this analogy, the case where the foliation has a dense leaf corresponds to the case where the algebraic scheme is flat over spec ℤ e.g. to spec ℤ itself. In this situation the foliation cohomology which in general is infinite dimensional is not of a topological but instead of a very analytic nature. This can also be seen from its description in terms of global differential forms which are harmonic along the leaves. An optimistic guess would be that for arithmetic schemes χ there exist foliated dynamical systems X whose reduced leafwise cohomology gives the desired cohomology of χ. If χ is an elliptic curve over a finite field this is indeed the case with X a generalized solenoid, not a manifold, [D3].
We illustrate this philosophy by comparing the “explicit formulas” in analytic number theory to a transversal index theorem.
We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma$ by automorphisms of compact abelian groups in terms of Fuglede–Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma$-action by means of a ‘fundamental homoclinic point’ and the description of entropy in terms of the renormalized logarithmic growth rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n,n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma$ with trivial intersection.
The Beilinson conjectures describe the leading coefficients of L-series of varieties over number fields up to rational factors in terms of generalized regulators. We begin with a short but almost selfcontained introduction to this circle of ideas. This is possible by using Bloch's description of Beilinson's motivic cohomology and regulator map in terms of higher Chow groups and generalized cycle maps. Here we follow [Bl3] rather closely. We will then sketch how much of the known evidence in favour of these conjectures — to the left of the central point — can be obtained in a uniform way. The basic construction is Beilinson's Eisenstein symbol which will be explained in some detail. Finally in an appendix a map is constructed from higher Chow theory to a suitable Ext-group in the category of mixed motives as defined by Deligne and Jannsen. This smooths the way towards an interpretation of Beilinson's conjectures in terms of a Deligne conjecture for critical mixed motives [Sc2]. It also explains how work of Harder [Ha2] and Anderson fits into the picture.
For further preliminary reading on the Beilinson conjectures, one should consult the Bourbaki seminar of Soulé [Sol], the survey article by Ramakrishnan [Ra2] and the introductory article by Schneider [Sch]. For the full story see the book [RSS] and of course Beilinson's original paper [Bel]. Here one will also find the conjectures for the central and near-central points, which for brevity we have omitted here.
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