On some implications of formalized theories in our life

The formalized theories in which are considered different types of logics give us an easier way of understanding of our own interpretations of the concepts and of the events of life

THE FULFILLED EUCLIDEAN PLANE

The fulfilled euclidean plane is the real projective plane, completed with the infinite point of its infinite line denoted new incidence structure. This is a structure with neighbouring elements, in which the unicity of the line through two distinct points is not assured. This new Geometry is a Smarandacheian structure introduced here which generalizes and unites in the same time: Euclid, Bolyai Lobacewski Gauss and Riemann Geometries

Integral canonical models of unitary Shimura varieties

We prove the existence of integral canonical models of unitary Shimura varieties in arbitrary unramified mixed characteristic. Errata to [Va1] are also included.Comment: 27 pages. Accepted (in final form) for publication in Asian J. Mat

Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type with respect to h--hyperspecial subgroups as pro-\'etale covers of N\'eron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.Comment: 87 pages. Final version, to appear in Mathematische Nachrichten (most alignment issues kept loose to match with the layout of the journal

Deformation subspaces of p-divisible groups as formal Lie groups associated to p-divisible groups

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$ which is not isoclinic. Let \scrD (resp. \scrD_k) be the formal deformation space of $D$ over \Spf(W(k)) (resp. over \Spf(k)). We use axioms to construct formal subschemes \scrG_k of \scrD_k that: (i) have canonical structures of formal Lie groups over \Spf(k) associated to $p$-divisible groups over $k$, and (ii) give birth, via all geometric points \Spf(K)\to\scrG_k, to $p$-divisible groups over $K$ that are isomorphic to $D_K$. We also identify when there exist formal subschemes \scrG of \scrD which lift \scrG_k and which have natural structures of formal Lie groups over \Spf(W(k)) associated to $p$-divisible groups over $W(k)$. Applications to Traverso (ultimate) stratifications are included as well.Comment: 36 pages. To appear in J. Alg. Geo

Surjectivity Criteria for p-adic Representations, Part I

We prove several surjectivity criteria for $p$-adic representations. In particular, we classify all adjoint and simply connected group schemes $G$ over the Witt ring $W(k)$ of a finite field $k$ such that the epimorphism $G(W_2(k))\twoheadrightarrow G(k)$ has a section.Comment: 28 page