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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.), 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.
We introduce a notion of F1 structures based on quasi-unipotent endomorphisms on homology. We also consider F1 structures based on the integral Bost–Connes algebra and its endomorphisms. In both cases we consider
lifts of these structures, via an equivariant Euler charactetristic, to the level of
Grothendieck rings and further lifts, via the formalism of assembler categories,
to homotopy theoretic spectra.