## 10 Citations

Segal’s Gamma rings and universal arithmetic

- Mathematics
- 2020

Segal's Gamma-rings provide a natural framework for absolute algebraic geometry. We use Almkvist's global Witt construction to explore the relation with J. Borger F1-geometry and compute the Witt…

BC-system, absolute cyclotomy and the quantized calculus

- Mathematics
- 2021

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the ”zeta sector” of the latter space as the Scaling Site. The new…

Some remarks on blueprints andF1-schemes

- Mathematics
- 2021

Over the past two decades several different approaches to defining a geometry over F1 have been proposed. In this paper, relying on Toën and Vaquié’s formalism (J.K-Theory 3: 437–500, 2009), we…

Riemann-Roch for Spec Z

- Mathematics
- 2022

We prove a Riemann-Roch formula for Arakelov divisors on Spec Z equating the integer valued Euler characteristic with a simple modiﬁca-tion of the traditional expression ( i.e. the degree of the…

1 SPECTRAL TRIPLES and ζ-CYCLES

- Mathematics
- 2021

We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S. We show both…

Spectral Triples and Zeta-Cycles

- Mathematics
- 2021

We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S. We show both…

Geometric morphisms between toposes of monoid actions: factorization systems

- Mathematics
- 2022

LetM, N be monoids, and PSh(M), PSh(N) their respective categories of right actions on sets. In this paper, we systematically investigate correspondences between properties of geometric morphisms…

Toric Hall algebras and infinite-dimensional Lie algebras

- Mathematics
- 2020

We associate to a projective $n$-dimensional toric variety $X_{\Delta}$ a pair of co-commutative (but generally non-commutative) Hopf algebras $H^{\alpha}_X, H^{T}_X$. These arise as Hall algebras of…

On Quiver Representations over $\mathbb {F}_{1}$

- MathematicsAlgebras and Representation Theory
- 2021

We study the category $\textrm{Rep}(Q,\mathbb{F}_1)$ of representations of a quiver $Q$ over "the field with one element", denoted by $\mathbb{F}_1$, and the Hall algebra of…

Some remarks on blueprints and $${\pmb {{\mathbb {F}}}_1}$$-schemes

- MathematicsSão Paulo Journal of Mathematical Sciences
- 2021

<jats:p>Over the past two decades several different approaches to defining a geometry over <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathbb F}_1}$$</jats:tex-math><mml:math…

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- Mathematics
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In this article we use the theories of relative algebraic geometry and of homotopical algebraic geometry (cf. [HAGII]) to construct some categories of schemes defined under Spec ℤ. We define the…

New Approach to Arakelov Geometry

- Mathematics
- 2007

This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct…

Equations of tropical varieties

- Mathematics
- 2016

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent…

Sheaves in geometry and logic: a first introduction to topos theory

- Mathematics
- 1992

This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various…

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- 1977

It’s better to think of Algebraic Geometry as indicating a sub-area of mathematics as a whole, rather than a very precisely defined subfield.

Schemes over 𝔽1 and zeta functions

- MathematicsCompositio Mathematica
- 2010

Abstract We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ $C=\overline {\mathrm {Spec}\,\Z }$ over 𝔽1, whose corresponding zeta function is the complete Riemann…

Number fields and function fields: two parallel worlds

- Mathematics
- 2005

* Preface * Participants * List of Contributors * G. Bockle: Arithmetic over Function Fields: A Cohomological Approach * T. van den Bogaart and B. Edixhoven: Algebraic Stacks Whose Number of Points…

Trace formula in noncommutative geometry and the zeros of the Riemann zeta function

- Mathematics
- 1998

Abstract. We give a spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric…