A taste of Weil theory in characteristic one

Thas, Koen

doi:10.4171/157-1/8

Cited by 2 publications

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“…In a number of works ( [6], [7] and [8]) on motives and regularized determinants, Deninger played with the possibility of translating Weil's proof of the Riemann Hypothesis for function fields of projective curves over finite fields F q to the hypothetical curve Spec(Z). This idea also occurred, for instance, in Haran [9], and circulated in work of Smirnov [15] -see [20]. In [7], Deninger gave a description of conditions on a certain category M of motives which might allow such a translation.…”

Section: Deninger-manin Theorymentioning

confidence: 99%

Counting points and acquiring flesh

Thas

2017

Innov. Incidence Geom.

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This set of notes is based on a lecture I gave at "50 years of Finite Geometry -A conference on the occasion of Jef Thas's 70th birthday," in November 2014. It consists essentially of three parts: in a first part, I introduce some ideas which are based in the combinatorial theory underlying F1, the field with one element. In a second part, I describe, in a nutshell, the fundamental scheme theory over F1 which was designed by Deitmar. The last part focuses on zeta functions of Deitmar schemes, and also presents more recent work done in this area.

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Section: Deninger-manin Theorymentioning

confidence: 99%

Counting points and acquiring flesh

Thas

2017

Innov. Incidence Geom.

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Projective spaces over

Thas

2018

J of Combinatorial Designs

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In this essay, we study various notions of projective space (and other schemes) over F 1 ℓ, with F 1 denoting the field with one element. Our leading motivation is the “Hidden Points Principle,” which shows a huge deviation between the set of rational points as closed points defined over F 1 ℓ and the set of rational points defined as morphisms monospace𝚂𝚙𝚎𝚌 ( double-struckF 1 ℓ ) ↦ scriptX. We also introduce, in the same vein as Kurokawa [Proc. Jpn. Acad. Ser. A Math. Sci. 81 (2005), pp. 180–184], schemes of F 1 ℓ‐type and consider their zeta functions.

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A taste of Weil theory in characteristic one

Abstract: Abstract. In this very short and sketchy chapter, we draw some pictures on the arithmetic theory of F1.

Cited by 2 publications

References 16 publications

Counting points and acquiring flesh

Counting points and acquiring flesh

Projective spaces over

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