By Koen Thas

It's been identified for it slow that geometries over finite fields, their automorphism teams and likely counting formulae regarding those geometries have fascinating guises while one we could the dimensions of the sphere visit 1. nevertheless, the nonexistent box with one point, F1

, provides itself as a ghost candidate for an absolute foundation in Algebraic Geometry to accomplish the Deninger–Manin software, which goals at fixing the classical Riemann Hypothesis.

This booklet, that's the 1st of its style within the F1

-world, covers a number of parts in F1

-theory, and is split into 4 major elements – Combinatorial concept, Homological Algebra, Algebraic Geometry and Absolute Arithmetic.

Topics handled comprise the combinatorial concept and geometry in the back of F1

, specific foundations, the mix of other scheme theories over F1

which are almost immediately on hand, causes and zeta features, the Habiro topology, Witt vectors and overall positivity, moduli operads, and on the finish, even a few arithmetic.

Each bankruptcy is punctiliously written by means of specialists, and in addition to elaborating on identified effects, fresh effects, open difficulties and conjectures also are met alongside the way.

The range of the contents, including the secret surrounding the sector with one point, may still allure any mathematician, despite speciality.

Keywords: the sphere with one point, F1

-geometry, combinatorial F1-geometry, non-additive class, Deitmar scheme, graph, monoid, intent, zeta functionality, automorphism team, blueprint, Euler attribute, K-theory, Grassmannian, Witt ring, noncommutative geometry, Witt vector, overall positivity, moduli house of curves, operad, torificiation, Absolute mathematics, counting functionality, Weil conjectures, Riemann speculation

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**Extra info for Absolute Arithmetic and F1-geometry**

**Example text**

Thas, Notes on F1 , I. Combinatorics of D0 -schemes and F1 -geometry. Unpublished notes, 2012. [17] K. Thas, The structure of Deitmar schemes, I. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), 21–26. [18] K. Thas, Projective spaces over F1 . Preprint. [19] K. Thas and D. Zagier, Finite projective planes, Fermat curves, and Gaussian periods. J. Eur. Math. Soc. (JEMS ) 10 (2008), 173–190. [20] J. Tits, Sur les analogues alg´ebriques des groupes semi-simples complexes. Centre Belge Rech. , Colloque d’Alg`ebre sup´erieure, Bruxelles du 19 au 22 d´ec.

Linear representations of Bn . 2 = γ / GLn (F1 ) ∼ = Sn , where passing from the first row to the second means passing to the limit q → 1. The first row should be seen as a class of arrows for varying q taking values in the set of prime powers. This is what Kapranov and Smirnov do in their manuscript [10]. 2 by GLn (F1 [X]), and also suggest that the evaluation morphism X = 0 yields γ. Their motivation is a theorem of Drinfeld which states that over a finite field Fq , the profinite completion of GLn (Fq [X]) is embedded in the fundamental group of the space of q-polynomials of degree n in a rather similar way Bn is 32 Koen Thas the fundamental group of M Cn [X].

Be/∼kthas/Fun, 2008. [14] Yu. Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa). Columbia University Number Theory Seminar (New York, 1992), Ast´erisque 228 (1995), 121–163. [15] C. Soul´e, Les vari´et´es sur le corps a ` un ´el´ement. Mosc. Math. J. 4 (2004), 217–244, 312. [16] K. Thas, Notes on F1 , I. Combinatorics of D0 -schemes and F1 -geometry. Unpublished notes, 2012. [17] K. Thas, The structure of Deitmar schemes, I. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), 21–26.