This is a continuation of [10], where it was shown that K-equivalent complex projective manifolds have the same Betti numbers by using the theory of p-adic integrals and Deligne’s solution to the Weil conjecture. The aim of this note is to show that with a little more book-keeping work, namely by applying Faltings’ p-adic Hodge Theory, our p-adic method also leads to the equivalence of Hodge nu...

In this paper, we investigate some properties for the q-tangent numbers and polynomials. By using these properties, we give some interesting identities on the q-tangent polynomials and Bernstein polynomials. Throughout this paper, let p be a fixed odd prime number. The symbol, Zp, Qp and Cp denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic ...

In this short survey we look at a few basic features of p-adic numbers, somewhat with the point of view of a classical analyst. In particular, with p-adic numbers one has arithmetic operations and a norm, just as for real or complex numbers. Let Z denote the integers, Q denote the rational numbers, R denote the real numbers, and C denote the complex numbers. Also let | · | denote the usual abso...

Let p be a fixed prime number. Throughout this paper, the symbols Z, Zp, Qp, and Cp denote the ring of rational integers, the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N ∪ {0}. The p-adic absolute value in Cp is normalized in such a way that |p|p 1/p see 1–17 . Fo...

Let p be a fixed prime. Throughout this paper Zp, Qp, C and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Qp, cf. [1], [3], [6], [10]. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = p. Kubota and Leopoldt proved the existence of meromorphic...

Let p be a fixed prime number. Throughout this paper, the symbols Z, Zp, Qp, and Cp denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure ofQp, respectively. LetN be the set of natural numbers andZ N∪{0}. Let νp be the normalized exponential valuation ofCp with |p|p p−νp p p−1. Let UD Zp be the space of u...

where ((x)) = x − [x]G − 1 2 , if x / ∈ Z, ((x)) = 0, x ∈ Z, where [x]G is the largest integer ≤ x cf. ([1], [5], [9], [11], [12], [13]). In this paper, Zp, Qp, Cp, C and Z, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, the p-adic completion of the algebraic closure of Qp normalized by |p|p = p −1, and the complex field and integer numbers. Let q be an ...

Abstract The p -adic completion ? of the rational numbers induces a different absolute value |?| than typical | ?| we have on real numbers. In this paper compare and contrast functions f : ? + ? , for which composition with p-adic metric d generated by is still usual preserving that preserve Euclidean ?. particular, it shown ? an ultrametric if only there function g such = every . Some general ...

Let p be a fixed prime number. Throughout this paper, the symbols Z, Zp, Qp, and Cp denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers. The normalized valuation in Cp is denoted by | · |p with |p|p 1/p. Let UD Zp be the space of uniformly differe...

Let p be a fixed odd prime. Throughout this paper, Zp,Qp,Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp. The p-adic absolute value | |p on Cp is normalized so that |p|p 1/p. Let Z>0 be the set of natural numbers and Z≥0 Z>0 ∪ {0}. As is well known, the Bernoulli polynomials Bn x are defined by the ge...