Fun with $\F_1$

By Alain Connes, C. Consani, Matilde Marcolli. 2008

55 citations recorded by the source corpus.

We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost–Connes naturally arises by extension of scalars from the “field with one element” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that “field,” while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra. Video. For a video summary of this paper, please visit http:// www.youtube.com/watch?v=az_0pxm1jrI.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/564db929e04d528c7a40a970622efbf62b6aebd0

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Citations in the corpus

Mapping F_1-land:An overview of geometries over the field with one element (2009, background): “Shortly after that, many different approaches toF1-geometry arose.”
Lambda-rings and the field with one element (2009, unknown): “For these approaches see the following references: Baez [3], Connes–Consani [13], Connes–Consani–Marcoli [14], Deitmar [19][18][20], Diers [26], Durov [27], Ha-ran [36] [35], Kapranov [38], Kapranov–Smirnov [39], Kurokawa [42][43][44], Kurokawa– Ochiai–Wakayama [45], Smirnov [55], Tate–Voloch [59],…”
Toric varieties, monoid schemes and $cdh$ descent (2011, background): “Assume k is a commutative regular noetherian ring containing an infinite field and let G be a presheaf of spectra defined on the category of schemes of finite type over k.”
Schemes over 𝔽1 and zeta functions (2009, methodology): “thetical curve C= SpecZis related to the interpretation of the explicit formulae as a trace formula using the noncommutative geometric framework of the ad`ele class space as developed in [2], [3] and [4]. The paper ends with two appendices. §7 presents a very simple characterization of monoids comin”