Noncommutative Geometry and Number Theory / Aspects of Mathematics (PDF)
Where Arithmetic meets Geometry and Physics
(Sprache: Englisch)
In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and...
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In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local Lfactors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.
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Archimedean cohomology revisited (S. 107-108)Caterina Consani and Matilde Marcolli
Abstract. Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy.
We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ".ber at arithmetic in.nity" of an arithmetic variety and the complex of nearby cycles in the geometry of a degeneration over a disk, and we recall Deningers approach to the archimedean cohomology through an interpretation as global sections of a analytic Rees sheaf. We show that action of the Lefschetz, the log of monodromy and the log of Frobenius on the archimedean cohomology combine to determine a spectral triple in the sense of Connes. The archimedean part of the Hasse-Weil L-function appears as a zeta function of this spectral triple. We also outline some formal analogies between this cohomological theory at arithmetic in.nity and Giventals homological geometry on loop spaces.
1. Introduction
C. Deninger produced a uni.ed description of the local factors at arithmetic in.nity and at the .nite places where the local Frobenius acts semi-simply, in the form of a RaySinger determinant of a "logarithm of Frobenius" Ö on an in.nite dimensional vector space (the archimedean cohomology H·ar(X) at the archimedean places, [11]). The .rst author gave a cohomological interpretation of the space H·ar(X), in terms of a double complex K·,· of real di.erential forms on a smooth projective algebraic variety X (over C or R), with Tate-twists and suitable cuto.s, together with an endomorphism N,
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which represents a "logarithm of the local monodromy at arithmetic infinity". Moreover, in this theory the cohomology of the complex Cone(N)· computes real Deligne cohomology of X, [9].
The construction of[9] is motivated by a dictionary of analogies between the geometry of the tubular neighborhoods of the "fibers at arithmetic in.nity" of an arithmetic variety X and the geometric theory of the limiting mixed Hodge structure of a degeneration over a disk. Thus, the formulation and notation used in [9] for the double complex and archimedean cohomology mimics the de.nition, in the geometric case, of a resolution of the complex of nearby cycles and its cohomology([28]). In Section 2 and 3 we give an equivalent description of Consanis double complex, which allows us to investigate further the structure induced on the complex and the archimedean cohomology by the operators N, Ö, and the Lefschetz operator L.
In Section 4 we illustrate the analogies between the complex and archimedean cohomology and a resolution of the complex of nearby cycles in the classical geometry of an analytic degeneration with normal crossings over a disk. In Section 5 we show that, using the ConnesKreimer formalism of renormalization, we can identify the endomorphism N with the residue of a Fuchsian connection, in analogy to the log of the monodromy in the geometric case. In Section 6 we recall Deningers approach to the archimedean cohomology through an interpretation as global sections of a real analytic Rees sheaf over R.
The construction of[9] is motivated by a dictionary of analogies between the geometry of the tubular neighborhoods of the "fibers at arithmetic in.nity" of an arithmetic variety X and the geometric theory of the limiting mixed Hodge structure of a degeneration over a disk. Thus, the formulation and notation used in [9] for the double complex and archimedean cohomology mimics the de.nition, in the geometric case, of a resolution of the complex of nearby cycles and its cohomology([28]). In Section 2 and 3 we give an equivalent description of Consanis double complex, which allows us to investigate further the structure induced on the complex and the archimedean cohomology by the operators N, Ö, and the Lefschetz operator L.
In Section 4 we illustrate the analogies between the complex and archimedean cohomology and a resolution of the complex of nearby cycles in the classical geometry of an analytic degeneration with normal crossings over a disk. In Section 5 we show that, using the ConnesKreimer formalism of renormalization, we can identify the endomorphism N with the residue of a Fuchsian connection, in analogy to the log of the monodromy in the geometric case. In Section 6 we recall Deningers approach to the archimedean cohomology through an interpretation as global sections of a real analytic Rees sheaf over R.
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Autoren-Porträt von Caterina Consani
Prof. Dr. Caterina Consani, Department of Mathematics, The Johns Hopkins University, Baltimore, USAProf. Dr. Matilde Marcolli, Max-Planck Institute for Mathematics, Bonn, Germany
Bibliographische Angaben
- Autor: Caterina Consani
- 2007, 2006, 372 Seiten, Englisch
- Herausgegeben: Caterina Consani, Matilde Marcolli
- Verlag: Vieweg+Teubner Verlag
- ISBN-10: 3834803529
- ISBN-13: 9783834803528
- Erscheinungsdatum: 18.12.2007
Abhängig von Bildschirmgröße und eingestellter Schriftgröße kann die Seitenzahl auf Ihrem Lesegerät variieren.
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