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 dictionary 11 3. Quantum statistical mechanics and arithmetic 12 3.1. The Bost–Connes endomotive 14 3.2. Scaling as Frobenius in characteristic zero 15 4. The adeles class space 16 4.1. Cyclic module 17 4.2. The restriction map 17 4.3. The Morita equivalence and cokernel for K = Q 19 4.4. The cokernel of ρ for general global fields 20 4.5. Trace pairing and vanishing 24 5. Primitive cohomology 25 6. A cohomological Lefschetz trace formula 27 6.1. Weil's explicit formula as a trace formula 27 6.2. Weil Positivity and the Riemann Hypothesis 28 7. Correspondences 29 7.1. The scaling correspondence as Frobenius 29 7.2. Fubini's theorem and the trivial correspondences 31 8. Thermodynamics and geometry of the primes 33 8.