Leseprobe
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Weil's Conjecture for Function Fields
Jacob Lurie, Dennis Gaitsgory
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil’s conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil’s conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ℓ-adic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors.
Using a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weil’s conjecture. The proof of the product formula will appear in a sequel volume.
Kundenbewertungen
Theory, Eigenvalues and eigenvectors, Multivariate statistics, Complex projective space, Commutative algebra, Mathematical theory, Quantification (science), Gaussian measure, Riemannian manifold, Ricci curvature, Theorem, Differential topology, Partial differential equation, Algebraic variety, Conjecture, Existential quantification, Renormalization, Analytic function, Cartesian product, Dirac equation, Hilbert space, Coordinate singularity, Intersection homology, Rational number, Variable (mathematics), Gaussian curvature, Quantum field theory, Dirac measure, Nonlinear Schrödinger equation, Gravitational constant, Riemannian geometry, Hodge theory, Requirement, Differential form, Function space, Transcendental number, Likelihood function, Lefschetz duality, Computation, Homology (mathematics), Algebraic topology (object), Cohomology ring, Pseudo-Riemannian manifold, Algebraic curve, Semisimple algebraic group, Equation, Differential geometry, Kähler manifold, Hyperbolic partial differential equation, Einstein field equations, Mathematical induction, Home range, Differentiable manifold, Holomorphic vector bundle, Analytic torsion, Cohomology, Commutative ring, Riemann surface, Yang–Mills theory, Minkowski space, Topological manifold, Functor, Curvature, Lefschetz hyperplane theorem, Projective variety, Maxwell's equations, Quotient space (topology), Ball (mathematics), Exponential map (Lie theory), Geometry