CYCLIC HOMOLOGY AND ARITHMETIC

Cyclic homology has recently revealed its potential in relation to the description of Serre's Archimedean local factors in the Hasse-Weil L-function of an arithmetic variety as shown in the paper by A. Connes and C. Consani :  Cyclic homology, Serre's local factors and the lambda-operations. The elaboration of this topic constitutes one of the two leading themes of the course that AC is developing at the Collège de France this year. Cyclic cohomology was introduced and widely publicized in 1981  as an essential tool in noncommutative differential geometry. The talk  Spectral sequence and homology of currents for operator algebras  given by AC at the 1981 Oberwolfach meeting introduced for instance the SBI long exact sequence and described the cyclic cohomology of the NC torus. In the context of algebraic geometry instead, the application of cyclic homology to schemes has a more recent evolution  and it has been mainly promoted in the work of C. Weibel.
 In number-theory, there are three fundamental sources of  L-functions 1) arithmetic varieties 2) geometric  Galois representations 3) automorphic representations.
To a smooth, projective variety X  defined over a number field K, corresponds its Hasse--Weil zeta function which is the alternate product of the factors $L(H^m,s)$ attached to the Galois representation on the $\ell$-adic \'etale cohomology $H^m(X_{\bar K},\Q_{\ell})$. The function $L(H^m,s)$ is defined as an infinite Euler product whose non-archimedean factors have an immediate geometric meaning  at places $\nu$ of K of good reduction for X (we assume here that $\ell$ does not divide $\nu$), by implementing the action of the geometric Frobenius on the étale cohomology  of the reduction of $X$ at $\nu$.
At an archimedean place v instead, the local L-factor is roughly a product of powers of (shifted) Gamma functions whose definition depends upon the Hodge structure on the m-th Betti cohomology of the complex  variety $X_{v}=X\times_K\C$.
The main result of the above paper states that the alternate product (as $m$ ranges from $0$ to twice the dimension of X) of Serre's Archimedean factors is the inverse graded determinant of the action on cyclic homology of $X$, with coefficients in infinite adeles $\prod_{v|\infty} K_v$, of the operator $(2\pi)^{-1}(s-\Theta)$, where $\Theta$ generates the lambda operations which are the analogue in cyclic homology of the Adams operations in algebraic K-theory.
 Several subtle features ought to be mentioned at this point. One of them is the nuance between cyclic homology and  archimedean cyclic homology (this is the one taken up in the above result) which parallels and reflects, in cyclic homology, the difference between reduced and unreduced real Deligne cohomology (we refer to the paper for more details).
This result promotes the development of the archimedean cyclic homology as a theory playing a natural role in the theory of  motives in algebraic geometry, in view of its connection to algebraic K-theory by the regulator maps. Moreover, it also suggests the study of a generalization of the above result at the non-archimedean places and the existence of a global Lefschetz formula in cyclic homology.
The second part of the course  will focus on the description of the Archimedean counterpart of the rings of periods in p-adic Hodge theory (especially $B_{cris}$ and $B_{dR}$), by elaborating on the results contained in the recent collaborative paper The universal thickening of the field of real numbers.   Fontaine's rings of p-adic periods play a fundamental role in arithmetic in view of the comparison theorems relating étale cohomology with coefficients in p-adic numbers, with the other fundamental cohomological theories of arithmetic varieties and in particular with the de-Rham cohomology thus realizing Grothendieck's  idea of ``foncteur mysterieux''.
At a real place, the transposition of the construction of the rings of p-adic periods yields non-trivial relevant rings endowed with a canonical one parameter group of automorphisms which replaces the Frobenius in Fontaine's construction. At a complex place, this construction produces fundamental algebraic structures whose applications  transcend the realm of arithmetic by producing a natural framework in which Feynman integrals in quantum field theory should be understood.
The class will be entirely given as blackboard-chalk talks.
AC and K. Consani.