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Published 2020 | Submitted
Book Section - Chapter Open

Homotopy types and geometries below Spec(ℤ)

Abstract

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.

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.

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Created:
August 19, 2023
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