Homotopy types and geometries below Spec Z

Yuri I. Manin; Matilde Marcolli

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arXiv · 1806.10801

Homotopy types and geometries below Spec Z

Yuri I. Manin·Matilde Marcolli

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Abstract

After the first heuristic ideas about `the field of one element' F_1 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.~To\"en and M.~Vaqui\'e, 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.

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Yuri I. Manin, Matilde Marcolli. 2018-06-28. Homotopy types and geometries below Spec Z. https://arxiv.org/abs/1806.10801

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