Segal's Gamma rings and universal arithmetic

Absolute algebra and Segal's Gamma sets

We show that the basic categorical concept of an S-algebra as derived from the theory of Segal's Gamma-sets provides a unifying description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoids, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry. The assembly map determines a functorial way to associate an S-algebra to a monad on pointed sets. The notion of an S-algebra is very familiar in algebraic topology where it also provides a suitabl…

On Absolute Algebraic Geometry, the affine case

We develop algebraic geometry for general Segal's Gamma-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under Spec Z). The starting observation is that the category obtained by gluing together the category of commutative rings and that of pointed commutative monoids, that we used in our previous work to define F1-schemes, is naturally a full subcategory of the category of Segal's Gamma-rings. In this paper we develop the affine case of this general algebraic geometry: one distinctive feature is that the spectrum Spec(A) of a Gamma-ring is…

（phi，Gamma）-模算术族的上同调

Cohomology of arithmetic families of (phi,Gamma)-modules

证明了（phi，Gamma）-模的算术族的（phi，Gamma）-上同调和Iwasawa上同调的有限性和与基变换的相容性.利用这个有限性定理，我们表明，一个家庭的伽罗瓦表示，是密集逐点细化的意义上的Mazur实际上是三角形作为一个家庭在一个大的子空间。在Coleman-Mazur特征曲线的情况下，我们确定在所有点的行为。

量子连接在卡拉比-叶猜想三重流形上的算术几何

Arithmetic geometry of quantum connections on Calabi-Yau $3$-folds

研究量子连接在卡拉比-叶三重流形上的算术结构，建立模p量子cohomology上的逆Cartier算子与Frobenius拉回的关系，提出关于更一般半正辛流形上量子连接的猜想。

New foundations for geometry

We shall describe a simple generalization of commutative rings. The category GR of such "rings", contains the ordinary commutative rings (fully faithfully), but also the "integers" and "residue field" at a real or complex place of a field ; the "field with one element" (the initial object of GR ); the "arithmetical surface" ( the sum in the category GR of the integers with them self: Z(x)Z ) . We shall show that this geometry "see" the real and complex places of a number field (there is an Ostrowski theorem that the valuation sub-GR of a number field correspond to the finite and infinite prime…