Adic Number Field Research Articles

On a [formula omitted]-adic interpolation function for the [formula omitted]-extension of the generalized Bernoulli polynomials and its derivative

In [T. Kim, S.H. Rim, Generalized Carlitz’s q -Bernoulli numbers in the p -adic number field, Adv. Stud. Contemp. Math. 2 (2000) 9–19], the new q -extension of Bernoulli polynomials and generalized Bernoulli numbers attached to χ were constructed by using p -adic invariant integral on Z p . In this paper we construct the new q -extension of generalized Bernoulli polynomials attached to χ due to author and derive the existence of a specific p -adic interpolation function which interpolates the q -extension of generalized Bernoulli polynomials at negative integers. Finally, we give the values of partial derivative for this function and investigate some properties which are related to this interpolation function.

Mumford Dendrograms

An effective $p$-adic encoding of dendrograms is presented through an explicit embedding into the Bruhat-Tits tree for a $p$-adic number field. This field depends on the number of children of a vertex and is a finite extension of the field of $p$-adic numbers. It is shown that fixing $p$-adic representatives of the residue field allows a natural way of encoding strings by identifying a given alphabet with such representatives. A simple $p$-adic hierarchic classification algorithm is derived for $p$-adic numbers, and is applied to strings over finite alphabets. Examples of DNA coding are presented and discussed. Finally, new geometric and combinatorial invariants of time series of $p$-adic dendrograms are developped.

Exact algorithms for $p$-adic fields and epsilon constant conjectures

We describe an algorithmic approach to prove or disprove several recent conjectures for epsilon constants of Galois extensions of $p$-adic fields and number fields. For this approach, we must develop various algorithms for computations in Galois extensions of $p$-adic fields which are of independent interest. Our algorithms for $p$-adic fields are based on existing algorithms for number fields and are exact in the sense that we do not need to consider approximations to $p$-adic numbers.

Indecomposability of ideals of [formula omitted]-adic number fields

Let K/ k be a Galois p-extension over a p -adic number field k and G 1 be the first ramification group. We show that any ideal A of K is absolutely oG -indecomposable if and only if the order | G 1| does not divide the different D K/k .

Time-inhomogeneous stochastic processes on the {$p$}-adic number field

The theory of stochastic differential equation on the field of $p$-adics is initiated by Kochubei. In this article, we will focus on a class of random walks with a certain integrability to develop the theory of stochastic analysis in a way similar to the existing theory of stochastic analysis based on the Brownian motion. In fact, for any random walk in the class, we can introduce the notion of stochastic integral with respect to the random walk and justify the existence of the solution of the stochastic differential equations based on the random walk, where the stochastic differential equations admit not only Lipschitz continuous path dependent coefficients but also continuous coefficients growing at most linearly with respect to $p$-adic norm. Finally, we will see an example of stochastic process which can be covered by Dirichlet space theory and obtained also by solving stochastic differential equation.

On the Galois Module Structure of Ideals and Rings of All Integers of [formula omitted]-Adic Number Fields

On the Galois Module Structure of Ideals and Rings of All Integers of [formula omitted]-Adic Number Fields

The associated orders of rings of integers in Lubin-Tate division fields over the $p$-adic number field

The associated orders of rings of integers in Lubin-Tate division fields over the $p$-adic number field

Generating functions for the numbers of abelian extensions of a local field

The aim of this paper is to give an explicit formula for the numbers of abelian extensions of a p {\mathbf {p}} -adic number field and to study the generating function of these numbers. More precisely, we give the number of abelian extensions with given degree and ramification index, and the number of abelian extensions with given degree of any local field of characteristic zero. Moreover, we give a concrete expression of a generating function for these last numbers.

Note on a p-block of a finite group with abelian defect group

Let G be a finite group and B be a p-block of G with an abelian defect group D and with a root b in CD(D). Let Dl=CD(T(b)) and S1=lf^ ι\ where T(V) is the inertial group of b in NG(D). In [9, Theorem 1] we showed that the number k(B) (resp. l(B)) of ordinary (resp. modular) irreducible characters in B is equal to that of ordinary (resp. modular) irreducible characters in Sλ. In this paper we continue our study to show further properties on the p-block B^ Let K be the algebraic closure of the />-adic number field, o the ring of local integres in K and F be the residue class field of o. For a subgroup H of G, we denote by Tr# the relative trace map from (FG) to the center Z(FG) of the group ring FG, where (FG)={x^FG\hx=xh for any h^H}. For a ^-subgroup Q of G, we denote by SQ the Brauer homomorphism from (FG) Q to FCG(Q). The following is the main result.

Note on a p-block of a finite group with abelian defect group

Let G be a finite group and B be a p-block of G with an abelian defect group D and with a root b in CD(D). Let Dl=CD(T(b)) and S1=lf^ ι where T(V) is the inertial group of b in NG(D). In [9, Theorem 1] we showed that the number k(B) (resp. l(B)) of ordinary (resp. modular) irreducible characters in B is equal to that of ordinary (resp. modular) irreducible characters in Sλ. In this paper we continue our study to show further properties on the p-block B^ Let K be the algebraic closure of the />-adic number field, o the ring of local integres in K and F be the residue class field of o. For a subgroup H of G, we denote by Tr# the relative trace map from (FG) to the center Z(FG) of the group ring FG, where (FG)={x^FG\hx=xh for any h^H}. For a ^-subgroup Q of G, we denote by SQ the Brauer homomorphism from (FG) Q to FCG(Q). The following is the main result.

Extensions of lattices overp-solvable groups

By PETER SCHMID Suppose G is a finite p-solvable group for some prime p. Let K be a finite extension of the p- adic number field 9 and k = R/r~R its residue class field. Let U, Vbe RG-lattices ( = R-free RG-modules of finite R-rank). The objective of this note is to record the following. Theorem. Ext~o (U, V) = 0 if either (i) K|174 is irreducible belonging to a block with cyclic defect group and K is unramified over •, or (ii) k| = k| is irreducible and the ramification index of K over 9 is less than p- 1 (and hence p odd). Statement (i) generalizes a result by Berman [11 who considered the case where G has a normal cyclic Sylow p-subgroup (and K = 9 It has been already noticed by Gudivok [31 that the result does not hold, in general, if K is ramified over 9 In a certain sense, both statements rely on the Fong-Swan theorem. In the first case we use the theory of blocks with cyclic defect groups and the fact that the Brauer tree is a star if G is p-solvable. The second statement follows from the uniqueness part of the lifting theorem in [61. Thus solvability is a crucial assumption. 1. Extensions of lattices. Let G be a finite group and let U, Vbe RG-lattices. Multiplication with 7r gives the exact sequence O-*V&V-,fz~O where f'= V/IrV~k| Since U is R- projective, the induced sequence 0--Hom R (U, V)&HomR (U, V)~Hom R (U, I7") ~0 is exact as well, and Extlo (U, V) = H 1 (G, Hom R (U, V)). Lemma. Ext~o(O,V)=O if and only if the residue class map HomRc(U, V)~ Homk6 (U, V) is epimorphic. P r o o f. Application of the long exact cohomology sequence yields the exact sequence 0 ~HomRc (U, V) ~ HomRo (U, V) --*Homko (0, ~2)-~Ext~o (U, V) Z*Ext~a (U, V) .... .

Vertices of ideals of a -adic number field II

Letkbe a-adic number field with the ring 0 of all integers ink, andKbe a finite normal extension with Galois groupG.∏denotes a prime element of the ringof all integers inK. Then, an ideal (∏a) ofis an 0G-module. E. Noether showed that ifK/kis tamely ramified,is a free 0G-module. A. Fröhlich generalized E. Noether’s theorem as follows:is relatively projective with respect to a subgroupSofGif and only ifS⊇G1whereG1is the first ramification group ofK/k. Now we define the vertexV(∏a) of (∏a) as the minimal normal subgroupSofGsuch that (∏a) is relatively projective with respect to a subgroupSofG(cf. § 1). Then, the above generalization by A. Fröhlich impliesV() =G1. In the previous paper, we provedG1⊇ V(∏a) ⊇ G2, (whereG2is the second ramification group ofK/k(cf. Theorem 5). Further, we dealt with the case whereG=G1is of orderp2, and proved that ifV(∏a) ≠G1thena≡ 1(p2) andt2≡ 1(p2) for the second ramification numbert2ofK/k(cf. Theorems 15 and 21). The purpose of this paper is to prove the similar theorem for the wildly ramified p-extension of degreepn(Theorem 7).

Vertices of ideals of a $mathfrak{P}$-adic number field

Vertices of ideals of a $mathfrak{P}$-adic number field

On the module structure of a p-extension over a -adic number field

Throughout this paper, let p be an odd prime. Let k be a -adic number field and be the ring of all integers in k. Let K/k be a finite totally ramified Galois p-extension of degree pn with the Galois group G. Clearly the ring of all integers in K is an [G]-module. In the previous paper [4], we studied [G]-module structure of in a cyclic totally ramified p-extension, and we have obtained the condition for to be an indecomposable [G]-module.

On the module structure in a cyclic extension over a -adic number field

Let p be a prime. Let k be a -adic number field and be the ring of all integers of k. Let K/k be a cyclic totally ramified extension of degree pn with Galois group G. Clealy the ring of all integers of K is an [G]-module, and the purpose of this paper is to give a necessary and sufficient condition for the [G]-module to be indecomposable.

Characterization of the simple components of the group algebras over the $p$ -adic number field

Characterization of the simple components of the group algebras over the $p$ -adic number field

ON THE SCHUR INDEX OF PROJECTIVE REPRESENTATIONS OF FINITE GROUPS

This work is a study of the set of Schur indices of absolutely irreducible projective representations of nilpotent groups over a field which is a finite extension of the rational number field or of the rational -adic number field . Bibliography: 13 titles.

On the cohomology groups of $\mathfrak{p}$-adic number fields

On the cohomology groups of $\mathfrak{p}$-adic number fields

Some Remarks concerning $p$ -adic Number Field

Some Remarks concerning $p$ -adic Number Field

On a certain group concerning the $p$-adic number field

On a certain group concerning the $p$-adic number field