Noncommutative geometry and motives: the thermodynamics of endomotives

TL;DR

This paper integrates motives, noncommutative geometry, and thermodynamics to interpret zeros of L-functions cohomologically, linking algebraic and quantum statistical systems, and extends to archimedean local factors.

Contribution

It introduces a novel cohomological framework connecting motives, noncommutative geometry, and thermodynamics to analyze zeros of L-functions.

Findings

Cohomological interpretation of zeros of L-functions via noncommutative spaces.

Construction of endomotives from algebraic varieties with Galois action.

Lefschetz formula for archimedean local L-factors.

Abstract

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.