\contentsline {title}{The Weil proof and the geometry of the adeles class space}{331}
\contentsline {author}{Alain Connes\unskip {} \and Caterina Consani\unskip {} \and Matilde Marcolli\unskip {}}{331}
\contentsline {section}{\numberline {1}Introduction}{333}
\contentsline {section}{\numberline {2}A look at the Weil proof}{336}
\contentsline {subsection}{\numberline {2.1}Correspondences and divisors}{338}
\contentsline {subsection}{\numberline {2.2}The explicit formula}{339}
\contentsline {subsection}{\numberline {2.3}Riemann--Roch and positivity}{341}
\contentsline {subsection}{\numberline {2.4}A tentative dictionary}{344}
\contentsline {section}{\numberline {3}Quantum statistical mechanics and arithmetic}{345}
\contentsline {subsection}{\numberline {3.1}The Bost--Connes endomotive}{347}
\contentsline {subsection}{\numberline {3.2}Scaling as Frobenius in characteristic zero}{348}
\contentsline {section}{\numberline {4}The adeles class space}{349}
\contentsline {subsection}{\numberline {4.1}Cyclic module}{350}
\contentsline {subsection}{\numberline {4.2}The restriction map}{351}
\contentsline {subsection}{\numberline {4.3}The Morita equivalence and cokernel for ${\@mathbb K}={\@mathbb Q}$}{353}
\contentsline {subsection}{\numberline {4.4}The cokernel of $\rho $ for general global fields}{355}
\contentsline {subsection}{\numberline {4.5}Trace pairing and vanishing}{359}
\contentsline {section}{\numberline {5}Primitive cohomology}{360}
\contentsline {section}{\numberline {6}A cohomological Lefschetz trace formula}{362}
\contentsline {subsection}{\numberline {6.1}Weil's explicit formula as a trace formula}{362}
\contentsline {subsection}{\numberline {6.2}Weil Positivity and the Riemann Hypothesis}{363}
\contentsline {section}{\numberline {7}Correspondences}{364}
\contentsline {subsection}{\numberline {7.1}The scaling correspondence as Frobenius}{365}
\contentsline {subsection}{\numberline {7.2}Fubini's theorem and the trivial correspondences}{367}
\contentsline {section}{\numberline {8}Thermodynamics and geometry of the primes}{368}
\contentsline {subsection}{\numberline {8.1}The global Morita equivalence}{370}
\contentsline {subsection}{\numberline {8.2}The valuation systems}{372}
\contentsline {subsection}{\numberline {8.3}The curve inside the adeles class space}{375}
\contentsline {subsection}{\numberline {8.4}The valuation systems for ${\@mathbb K}={\@mathbb Q}$}{376}
\contentsline {subsection}{\numberline {8.5}The cyclic covering $\mathaccentV {tilde}07E\Xi _{{\@mathbb Q}}$ of $\Xi _{{\@mathbb Q}}$}{383}
\contentsline {subsection}{\numberline {8.6}Arithmetic subalgebra, Frobenius and monodromy}{384}
\contentsline {section}{\numberline {9}Functoriality of the adeles class space}{386}
\contentsline {subsection}{\numberline {9.1}The norm map}{387}
\contentsline {subsection}{\numberline {9.2}The Weil group and the transfer map}{388}
\contentsline {subsection}{\numberline {9.3}The covering}{388}
\contentsline {subsection}{\numberline {9.4}The function field case}{390}
\contentsline {section}{\numberline {10}Vanishing cycles: an analogy}{391}
\contentsline {subsection}{\numberline {10.1}Two real places}{392}
\contentsline {subsection}{\numberline {10.2}A real and a non-archimedean place}{394}
\contentsline {subsection}{\numberline {10.3}Singularities of maps}{396}
\contentsline {section}{References}{397}