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Noncommutative geometry and motives: the thermodynamics of endomotives

Connes, Alain and Consani, Caterina and Marcolli, Matilde (2007) Noncommutative geometry and motives: the thermodynamics of endomotives. Advances in Mathematics, 214 (2). pp. 761-831. ISSN 0001-8708. doi:10.1016/j.aim.2007.03.006.

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We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ℓ-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost–Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.

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Additional Information:© 2007 Elsevier Inc. Received 22 December 2005; accepted 22 March 2007. Available online 30 March 2007. This research was partially supported by the third author’s Sofya Kovalevskaya Award and by the second author’s NSERC grant 7024520. Part of this work was done during a visit of the first and third authors to the Kavli Institute in Santa Barbara, supported in part by the National Science Foundation under Grant No. PHY99-07949, and during a visit of the first two authors to the Max Planck Institute.
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Sofya Kovalevskaya AwardUNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)7024520
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13550
Deposited By: Tony Diaz
Deposited On:08 May 2009 17:53
Last Modified:08 Nov 2021 22:38

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