We study and investigate the p-adic arithmetic along with analysis of exact solution of linear equation systems over rational numbers. Initially we study the basic concepts involving the p-adic numbers and why they form a better representation. After that we describe a parallel implementation of an algorithm for solving systems of linear equations over the field of rational number based on the ...

We study a positive characteristic analog of the nonabelian Hodge structure constructed by Katzarkov, Pantev, and Toen on the homotopy type of a complex algebraic variety. Given a proper smooth scheme X over a perfect field of characteristic p and a Tannakian category C of isocrystals on X, we construct an object XC in a suitable homotopy category of simplicial presheaves whose category of loca...

We give an answer to the question as to whether quantifier elimination is possible in some infinite algebraic extensions of Qp (‘infinite p-adic fields’) using a natural language extension. The present paper deals with those infinite p-adic fields which admit only tamely ramified algebraic extensions (so-called tame fields). In the case of tame fields whose residue fields satisfy Kaplansky’s co...

Let E be an elliptic curve defined over Q. The aim of this paper is to make it possible to compute Heegner L-functions and anticyclotomic Λ-adic regulators of E, which were studied by Mazur-Rubin and Howard. We generalize results of Cohen and Watkins and thereby compute Heegner points of nonfundamental discriminant. We then prove a relationship between the denominator of a point of E defined ov...

Various measures of security of stream ciphers have been studied that are based on the problem of finding a minimum size generator for the keystream in some special class of generators. These include linear and p-adic spans, as well as π-adic span, which is based on a choice of an element π in a finite extension of the integers. The corresponding sequence generators are known as linear feedback...

The aim of this article is to present a new proof of a theorem of Karl Rubin (see [Ru] and Thm. 1 below) relating values of the Katz p-adic L-function of an imaginary quadratic field at certain points outside its range of classical interpolation to the formal group logarithms of rational points on CM elliptic curves. This theorem has been seminal in providing a motivation for Perrin-Riou’s form...

In the recent, many mathematicians studied the multiple zeta function in the complex number field. In this paper we construct the p-adic analogue of multiple zeta function which interpolates the generalized multiple Bernoulli numbers attached to χ at negative integers. §1. Introduction Let p be a fixed prime. Throughout this paper Z p , Q p , C and C p will, respectively, denote the ring of p-a...

Let E be an elliptic curve defined over Q. The aim of this paper is to make it possible to compute Heegner L-functions and anticyclotomic Λ-adic regulators of E, which were studied by Mazur-Rubin and Howard. We generalize results of Cohen and Watkins, which enable us to compute Heegner points of non-fundamental discriminant. We then prove a relationship between the denominator of a point of E d...

Part I. First steps in p-adic Hodge theory 4 1. Motivation 4 1.1. Tate modules 4 1.2. Galois lattices and Galois deformations 6 1.3. Aims of p-adic Hodge theory 7 1.4. Exercises 9 2. Hodge–Tate representations 10 2.1. Basic properties of CK 11 2.2. Theorems of Tate–Sen and Faltings 12 2.3. Hodge–Tate decomposition 15 2.4. Formalism of Hodge–Tate representations 17 2.5. Exercises 24 3. Étale φ-m...

In this paper, we continue our study of the issue of the extent to which a hyperbolic curve over a finite extension of the field of p-adic numbers is determined by the profinite group structure of its étale fundamental group. Our main results are that: (i) the theory of correspondences of the curve — in particular, its arithmeticity — is completely determined by its fundamental group; (ii) when...