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.

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.

scholarly journals Formal Group Laws and Chromatic Symmetric Functions of Hypergraphs

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Power Series ◽  
Symmetric Functions ◽  
Formal Group ◽  
Combinatorial Interpretation ◽  
Recursive Structure ◽  
Combinatorial Objects ◽  
Formal Group Law ◽  
International Audience ◽  
Formal Group Laws ◽  
Group Law

International audience If $f(x)$ is an invertible power series we may form the symmetric function $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ which is called a formal group law. We give a number of examples of power series $f(x)$ that are ordinary generating functions for combinatorial objects with a recursive structure, each of which is associated with a certain hypergraph. In each case, we show that the corresponding formal group law is the sum of the chromatic symmetric functions of these hypergraphs by finding a combinatorial interpretation for $f^{-1}(x)$. We conjecture that the chromatic symmetric functions arising in this way are Schur-positive. Si $f(x)$ est une série entière inversible, nous pouvons former la fonction symétrique $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ que nous appelons une loi de groupe formel. Nous donnons plusieurs exemples de séries entières $f(x)$ qui sont séries génératrices ordinaires pour des objets combinatoires avec une structure récursive, chacune desquelles est associée à un certain hypergraphe. Dans chaque cas, nous donnons une interprétation combinatoire à $f^{-1}(x)$, ce qui nous permet de montrer que la loi de groupe formel correspondante est la somme des fonctions symétriques chromatiques de ces hypergraphes. Nous conjecturons que les fonctions symétriques chromatiques apparaissant de cette manière sont Schur-positives.

Stable cohomotopy and cobordism of abelian groups

1981 ◽  
Vol 90 (2) ◽  
pp. 273-278 ◽  
Author(s):  
C. T. Stretch
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Formal Group ◽  
Characteristic Class ◽  
Burnside Ring ◽  
Classifying Space ◽  
Formal Group Law ◽  
Stable Cohomotopy ◽  
Group Law

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

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.

On Ideals Annihilating the Toral Class of BP*((BZ/PK)N)

1993 ◽  
Vol 36 (3) ◽  
pp. 332-343 ◽  
Author(s):  
George Nakos
Keyword(s):  
Formal Group ◽  
Formal Group Law ◽  
Group Law

AbstractA sequence of ideals Ik,n ⊆ BP* is introduced, with the property: Ik,n ⊆ Ann(γk,n), where γk,n is the toral class of the Brown-Peterson homology of the n-fold product BZ/pk × ··· × BZ/pk. These ideals seem to play an interesting and yet unclear role in understanding Ann(γk,n). They are defined by using the formal group law of the Brown-Peterson spectrum BP, and some of their elementary properties are established. By using classical theorems of Landweber and of Ravenel-Wilson, the author computes the radicals of Ik,n and Ann(γk,n), and discusses a few examples.

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