  We combine aspects of the theory of motives in algebraic geometry with
noncommutative geometry and the classification of factors to obtain a
cohomological interpretation of the spectral realization of zeros of
$L$-functions. The analogue in characteristic zero of the action of the
Frobenius on l-adic cohomology is the action of the scaling group on the cyclic
homology of the cokernel (in a suitable category of motives) of a restriction
map of noncommutative spaces. The latter is obtained through the thermodynamics
of the quantum statistical system associated to an endomotive (a noncommutative
generalization of Artin motives). Semigroups of endomorphisms of algebraic
varieties give rise canonically to such endomotives, with an action of the
absolute Galois group. The semigroup of endomorphisms of the multiplicative
group yields the Bost-Connes system, from which one obtains, through the above
procedure, the desired cohomological interpretation of the zeros of the Riemann
zeta function. In the last section we also give a Lefschetz formula for the
archimedean local L-factors of arithmetic varieties.
