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 “fiber at arithmetic infinity” of an arithmetic variety and the complex of nearby cycles in the geometry of a degeneration over a disk, and we recall Deninger’s 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 infinity and Givental’s homological geometry on loop spaces.
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© 2006 Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden
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Consani, C., Marcolli, M. (2006). Archimedean cohomology revisited. In: Consani, C., Marcolli, M. (eds) Noncommutative Geometry and Number Theory. Aspects of Mathematics. Vieweg. https://doi.org/10.1007/978-3-8348-0352-8_5
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DOI: https://doi.org/10.1007/978-3-8348-0352-8_5
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