GROUP ACTION ON THE DEFORMATIONS OF A FORMAL GROUP OVER THE RING OF WITT VECTORS

2017 ◽  
Vol 235 ◽  
pp. 42-57
Author(s):  
OLEG DEMCHENKO ◽  
ALEXANDER GUREVICH
Keyword(s):  
Finite Field ◽  
Coordinate System ◽  
Group Action ◽  
Formal Group ◽  
Explicit Construction ◽  
Formal Groups ◽  
One Dimensional ◽  
Witt Vectors ◽  
Dieudonne Theory ◽  
Explicit Relation

A recent result by the authors gives an explicit construction for a universal deformation of a formal group $\unicode[STIX]{x1D6F7}$ of finite height over a finite field $k$ . This provides in particular a parametrization of the set of deformations of $\unicode[STIX]{x1D6F7}$ over the ring ${\mathcal{O}}$ of Witt vectors over $k$ . Another parametrization of the same set can be obtained through the Dieudonné theory. We find an explicit relation between these parameterizations. As a consequence, we obtain an explicit expression for the action of $\text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ on the set of ${\mathcal{O}}$ -deformations of $\unicode[STIX]{x1D6F7}$ in the coordinate system defined by the universal deformation. This generalizes a formula of Gross and Hopkins and the authors’ result for one-dimensional formal groups.

Kernels in the Category of Formal Group Laws

2016 ◽  
Vol 68 (2) ◽  
pp. 334-360 ◽  
Author(s):  
Oleg Demchenko ◽  
Alexander Gurevich
Keyword(s):  
Finite Field ◽  
Formal Group ◽  
Explicit Construction ◽  
Formal Groups ◽  
Witt Vectors ◽  
Formal Group Law ◽  
Formal Group Laws ◽  
Strict Isomorphism ◽  
Dieudonné Module ◽  
Group Law

AbstractFontaine described the category of formal groups over the ring of Witt vectors over a finite field of characteristic p with the aid of triples consisting of the module of logarithms, the Dieudonné module, and the morphism from the former to the latter. We propose an explicit construction for the kernels in this category in term of Fontaine's triples. The construction is applied to the formal norm homomorphism in the case of an unramified extension of ℚp and of a totally ramiûed extension of degree less or equal than p. A similar consideration applied to a global extension allows us to establish the existence of a strict isomorphism between the formal norm torus and a formal group law coming from L-series.

scholarly journals On explicit reciprocity law over formal groups

2004 ◽  
Vol 2004 (12) ◽  
pp. 607-635 ◽  
Author(s):  
Shaowei Zhang
Keyword(s):  
Formal Group ◽  
Tempered Distributions ◽  
Formal Groups ◽  
One Dimensional ◽  
Reciprocity Law ◽  
Explicit Reciprocity Law ◽  
Tate Formal Group

Let𝔣be a one-dimensional Lubin-Tate formal group overℤp. Colmez (1998) and Perrin-Riou (1994) proved an explicit reciprocity law for tempered distributions over the formal group𝔾m. In this paper, the general explicit reciprocity law over the formal group𝔣is proved.

scholarly journals Formal groups and Z -entropies

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160143 ◽  
Author(s):  
Piergiulio Tempesta
Keyword(s):  
Algebraic Topology ◽  
Formal Group ◽  
Point Of View ◽  
Trace Form ◽  
Formal Groups ◽  
Form Class ◽  
New Family ◽  
Mathematical Properties ◽  
Composition Process ◽  
Group Law

We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the Z-entropies . Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that every Z -entropy is composable (Tempesta 2016 Ann. Phys. 365 , 180–197. ( doi:10.1016/j.aop.2015.08.013 )). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon–Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the Z -entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

scholarly journals A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

2019 ◽  
Vol 31 (5) ◽  
pp. 1317-1330
Author(s):  
Russell Ricks
Keyword(s):  
Symmetric Space ◽  
Group Action ◽  
Spherical Building ◽  
Rigidity Result ◽  
One Dimensional ◽  
Higher Rank ◽  
Geodesically Complete ◽  
Euclidean Building ◽  
Alternate Condition

AbstractWe prove the following rank rigidity result for proper {\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of {\partial X} equals π and Γ does not act minimally on {\partial X}, then {\partial X} is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of {\partial X}, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

K3 of truncated polynomial rings over fields of characteristic two

1987 ◽  
Vol 101 (3) ◽  
pp. 509-521 ◽  
Author(s):  
Janet Aisbett ◽  
Victor Snaith
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Witt Vectors ◽  
Characteristic Two

Write F for the finite field, , having 2m elements. Let W2(F) denote the Witt vectors of length two over F (for a definition, see [4] or [10], §10). Write F(q) for the truncated polynomial ring, F[t]/(tq).

Correlation Technique for Estimating Traffic Speed from Cameras

2003 ◽  
Vol 1855 (1) ◽  
pp. 66-73 ◽  
Author(s):  
Todd N. Schoepflin ◽  
Daniel J. Dailey
Keyword(s):  
Coordinate System ◽  
Weather Conditions ◽  
Dimensional Subspace ◽  
Correlation Technique ◽  
Camera Model ◽  
One Dimensional ◽  
Congested Traffic ◽  
Adverse Weather Conditions ◽  
Adverse Weather ◽  
Favorable Weather

A new algorithm is presented for estimating speed from roadside cameras in uncongested traffic, congested traffic, favorable weather conditions, and adverse weather conditions. Individual vehicle lanes are identified and horizontal vehicle features are emphasized by using a gradient operator. The features are projected into a one-dimensional subspace and transformed into a linear coordinate system by using a simple camera model. A correlation technique is used to summarize the movement of features through a group of images and estimate mean speed for each lane of vehicles.

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