Christopher Deninger has elaborated a certain infinite-dimensional cohomological formalism which should provide a unifying approach to numerous zeta-functions, such as the zeta-function of a variety defined over a finite field, the Riemann zeta function, the Dedekind zeta function, the zeta-function of an arithmetic variety, the Dirichlet \(L\)-functions, and the motivic \(L\)-functions \(L(M,s)\) where \(M\) is a motif over a number field. At the moment this formalism is proved to exist only in very special cases.
In Section 2 of this paper the author considers the case of an elliptic curve \(E_0\) over a finite field. Admitting Hasse’s theorem he recalls Deninger’s construction of the foliated laminated Riemannian space \((S(E_0),{\mathcal F},g,\phi^t)\) endowed with an action of a flow \(\phi^t\). The cohomology groups appear here as \(L^2-\)leafwise cohomology groups and the author shows in detail how this cohomological formalism works in this case. He recalls Deninger’s proof of the fact that the Lefschetz trace formula for \(\phi^t\) acting on these cohomology groups is identical with the explicit formula for \(\zeta_{E_0}\). He views the foliated space \((S(E_0),{\mathcal F},\phi^t)\) as a noncommutative analogue of the set of geometric points \((\bar{E_0},\text{Fr})\), \((\phi^t)\) playing the role of the continuous version of the Frobenius endomorphism. In Section 3 he establishes a connection with Connes’ result applied to \(E_0\).
In Section 4 he considers a smooth proper absolutely irreducible variety \(Y\) over a finite field and he reviews Deninger’s non natural construction of an infinite-dimensional cohomology satisfying Poincaré duality and allowing to prove only the expected cohomological formula and the functional equation for \(\zeta_Y\). The underlying foliated space \((S_Y,{\mathcal F},g,\phi^t)\) which should allow to construct this cohomology in a natural way is not known to exist. He states a precise geometric hypothesis that such a foliated space \((S_Y,{\mathcal F},g,\phi^t)\) (and the flow \(\phi^t\) acting on \(S_Y\)) should satisfy in order to provide a natural construction of the expected cohomology. He shows, following Deninger, how this geometric hypothesis provides a new proof of all the expected properties for \(\zeta_Y\) including the Riemann hypothesis (i.e., Deligne’s Theorem).
In Section 5 he extracts from several papers of Deninger about the Riemann zeta-function \(\hat{\zeta}\), a conjecture stating the existence of a Riemannian laminated foliated space \((\bar{S_\mathbb Q},{\mathcal F})\) endowed with a flow \(\phi^t\) and satisfying certain geometric properties. He establishes a brief comparison between this conjectural foliated space and the noncommutative space constructed by Connes’ for \(\hat{\zeta}\). He explains how this conjecture implies the cohomological formula for \(\hat{\zeta}\) and the Riemann hypothesis.
For the entire collection see [Zbl 1075.58001].