Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields

Let k be a subfield of a p-adic field of odd residue characteristic, and let be the function field of a variety of dimension n1 over k. Then Hilbert's Tenth Problem for is undecidable. In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension 1 is undecidable.     

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Zp. This is done by defining an index preserving morphism which splits , where is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over .     

On nonlinear equations of p-adic strings for scalar tachyon fields

We consider boundary value problems for open and closed p-adic strings for scalar tachyon fields. Estimates for solutions to these problems and possible ways of constructing these solutions are obtained by reducing the problems to linear parabolic equations with nonlinear boundary conditions. We give an application of Gauss-type quadrature formulas to the numerical solution of the boundary value problems, and discuss the possibility of using these methods in multidimensional problems (d = 2). Original Russian Text ? V.S. Vladimirov, 2009, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 265, pp. 254–272.     

p-Adic ideals of p-rank d and the p-adic Nullstellensatz

In this paper we introduce and we study the notion of p-adic ideal of p-rank d and the one of p-adic radical of p-rank d of an ideal, by analogy with the notion of real ideal and the one of real radical of an ideal in the real case. Using those two notions and following more closely the real approach of Bochnak et al. (Géométrie algébrique réelle, Springer, Berlin 1987) we give a new proof of the p-adic Nullstellensatz.     

Central units in p-adic group rings

If p is a natural prime and G any p-group then an exact sequence is constructed to describe the group of units in the center of the p-adic group ring _p [G].     

Representation theory of p-adic groups and canonical bases

In this paper, we interpret the Gindikin–Karpelevich formula and the Casselman–Shalika formula as sums over Kashiwara–Lusztig?s canonical bases, generalizing the results of Bump and Nakasuji (2010) [7] to arbitrary split reductive groups. We also rewrite formulas for spherical vectors and zonal spherical functions in terms of canonical bases. In a subsequent paper Kim and Lee (preprint) [14], we will generalize these formulas to p-adic affine Kac–Moody groups.     

Lefschetz Formulae for p -Adic Groups

Abstract In this paper, Lefschetz formulae for torus actions on p-adic groups are proven. These are similar to comparable formulae for real Lie groups. Applications lie in the realm of dynamical zeta functions.     

Efficient p-adic Cell Decompositions for Univariate Polynomials

Cell decompositions are constructed for polynomials f(x)Z_p[x] of degree n, such that n<p, using O(n²) cells. When f is square-free this yields a polynomial-time algorithm for counting and approximating roots in Z_p. These results extend to give a polynomial-time algorithm in the bit model for fZ[x].     

Free-by-finite pro-p groups and integral p-adic representations

Let F be a free pro-p group of finite rank n and C_p^r{C_{p^r}} a cyclic group of order p ^r . In this work we classify p-adic representations C_p^r? GL_n(\mathbbZ_p){ C_{p^r}\longrightarrow GL_n(\mathbb{Z}_{p})} that can be obtained as a composite of an embedding C_p^r? Aut(F){C_{p^r}\longrightarrow {\rm Aut}(F)} with the natural epimorphism Aut(F)? GL_n(\mathbbZ_p){{\rm Aut}(F)\longrightarrow GL_n(\mathbb{Z}_{p})} .     

-Adic Dirichlet characters of algebraic function fields

A theory of precyclic P-extensions is developed which contains the well-known Witt theory. This theory makes it possible to describe so-called P-adic Dirichlet characters of function fields. In particular, for a trigonometric sum of the form where P is a prime number and f is a polynomial, its expression in terms of the zeros of a certain L function of the base field is obtainedTranslated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 97–129, 1975.     

On minimal decomposition of p-adic polynomial dynamical systems

A polynomial of degree ?2 with coefficients in the ring of p-adic numbers Zp is studied as a dynamical system on Zp. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on Z₂, we exhibit all its minimal subsystems.     

The Group Ring of SL2(p) over p-adic Integers for p Odd

Let p > 2 be a prime, R =  _p[ζ_p^f − 1], K =  _p[ζ_p^f − 1], and G = SL₂(p^f). The group ring RG is calculated nearly up to Morita equivalence: The projections of RG into the simple components of KG are given explicitly and the endomorphism rings and homomorphism bimodules between the projective indecomposable RG-lattices are described.     

On the chaotic behavior of a generalized logistic p-adic dynamical system

In the paper we describe basin of attraction p-adic dynamical system G(x)=(ax)²(x+1). Moreover, we also describe the Siegel discs of the system, since the structure of the orbits of the system is related to the geometry of the p-adic Siegel discs.     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

Codes over p-adic Numbers and Finite Rings Invariant under the Full Affine Group

Codes over p-adic numbers and over integers modulo p^d of block length p^m invariant under the full affine group AGL_m(F_p) are described.     

The Group Ring ofSL2(p) over thep-adic Integers

This paper describes the ring-theoretic structure of the group rings ofSL₂(p²) over thep-adic integers.     

Pro-P galois groups of function fields over local fields

For non-archimedean local field K and a prime number p we compute the finitely generated pro-p (closed) subgroups of the absolute Galois group of K(t). In addition, we characterize the finitely generated pro-p groups which occur as the maximal pro-p Galois group of algebraic extensions of K(t) containing a primitive pth root of unity.     

An algebroid function and its derivative

In this paper, we investigate the value distribution of an algebroid function and its derivative, and obtain two inequations between Nevanlinna characteristic function of an algebroid function and that of its derivative. We extend Chuang Chitai's theorem of meromorphic functions to algebroid functions.     

Chains of function spaces over Euclidian spaces and local fields

The Lipschitz class Lip(K, α) on a local field K is defined in [10], and an equivalent relationship between the Ho&#168;lder type space Cα(K)[9] and Lip(K,α) is given. In this note, we give a “chain of function spaces“ over Euclidian space by defining higher order continuous modulus in R, and point out that there is no need of higher order continuous modulus for describing the chain of function spaces over local fields.