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Homotopy types and geometries below Spec(ℤ)

Manin, Yuri I. and Marcolli, Matilde (2020) Homotopy types and geometries below Spec(ℤ). In: Dynamics: Topology and Numbers. Contemporary Mathematics. No.744. American Mathematical Society , Providence, RI, pp. 27-56. ISBN 978-1-4704-5100-4.

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After the first heuristic ideas about 'the field of one element' F₁ and 'geometry in characteristics 1' (J. Tits, C. Deninger, M. Kapranov, A. Smirnov et al.), there were developed several general approaches to the construction of 'geometries below Spec Z'. Homotopy theory and the 'the brave new algebra' were taking more and more important places in these developments, systematically explored by B. Toën and M. Vaquié, among others. This article contains a brief survey and some new results on counting problems in this context, including various approaches to zeta--functions and generalised scissors congruences. The new version includes considerable extensions and revisions suggested by I. Zakharevich.

Item Type:Book Section
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Additional Information:© 2020 American Mathematical Society. The second named author is partially supported by NSF grant DMS-1707882, and by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593. We thank Inna Zakharevich for several useful suggestions that were incorporated in the final version of the article.
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)RGPIN-2018-04937
Natural Sciences and Engineering Research Council of Canada (NSERC)RGPAS-2018-522593
Subject Keywords:Weil numbers, Bost–Connes systems, zeta functions, motivic integration, geometries below Spec Z, assemblers
Series Name:Contemporary Mathematics
Issue or Number:744
Classification Code:2010 Mathematics Subject Classification: 11M26, 14G40, 14G15
Record Number:CaltechAUTHORS:20200918-070725967
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:105442
Deposited By: Tony Diaz
Deposited On:18 Sep 2020 15:42
Last Modified:16 Nov 2021 18:43

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