The Weil Proof and the Geometry of the Adeles Class Space Alain Connes, Caterina Consani, and Matilde Marcolli

The Weil Proof and the Geometry of the Adeles Class Space

Dedicated to Yuri Manin on the occasion of his 70th birthday O simili o dissimili che sieno questi mondi non con minor raggione sarebe bene a l'uno l'essere che a l'altro Giordano Bruno – De l'infinito, universo e mondi Contents 1. Introduction 2 2. A look at the Weil proof 4 2.1. Correspondences and divisors 6 2.2. The explicit formula 8 2.3. Riemann–Roch and positivity 9 2.4. A tentative dict...

New perspectives in Arakelov geometry

In this paper we give a uni ed description of the archimedean and the totally split degenerate bers of an arithmetic surface, using operator algebras and Connes' theory of spectral triples in noncommutative geometry.

Quantum Statistical Mechanics over Function Fields Caterina Consani and Matilde Marcolli

It has become increasingly evident, starting from the seminal paper of Bost and Connes [3] and continuing with several more recent developments ([8], [10], [12], [13], [22], [24]), that there is a rich interplay between quantum statistical mechanics and arithmetic. In the case of number fields, the symmetries and equilibrium states of the Bost–Connes system are closely linked to the explicit cl...

Noncommutative Geometry, Quantum Fields and Motives

From Physics to Number theory via Noncommutative Geometry