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, 4, 6, 10]. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = p. When one talks of q-extension, q is variously considered as an...

Let p be a fixed prime number. 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. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. When one talks of q-extension, q is variously considered as an indeterminate,...

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rationalnumbers, the complex number field, and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. When one talks about q-extension, q is variously considered as an i...

Let F be a totally real field and χ an abelian totally odd character of F . In 1988, Gross stated a p-adic analogue of Stark’s conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross’s conjecture when F is a real quadratic field and χ is a narrow ring class ...

Many mathematicians have studied Euler numbers and Euler polynomials( see [1-11]). Euler polynomials posses many interesting properties and arising in many areas of mathematics and physics. In this paper we introduce the generalized q-Euler numbers and polynomials with weak weight α. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q de...

To Dong-Sik Kim on his 62nd birthday We try to generalize an explicit construction for an orthonormal system of eigen-functions of the Vladimirov operator on L 2 (Q n p) which was originally given by Vladimirov (1988). Also the multiwavelet analysis can be considered as the p-adic spectral analysis in L 2 (Q n p), using in part the recent work of Kozyrev (2002). denote the ring of integers, the...

Let p be a fixed prime number. Throughout this paper, Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integer, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp. Let N be the set of natural numbers and Z {0} ∪ N. Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1. When one talks of q-exte...

Fix an odd prime p. Let G be a compact p-adic Lie group containing a closed, normal, pro-p subgroup H which is abelian and such that G/H is isomorphic to the additive group of p-adic integers Zp. First we assume that H is finite and compute the Whitehead group of the Iwasawa algebra, Λ(G), of G. We also prove some results about certain localisation of Λ(G) needed in Iwasawa theory. Let F be a t...

This paper is about computational and theoretical questions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions Kv at the places v of K dividing p, a single quantity: the value of the p-adic modular form E2 associated to the elliptic curve. Thanks to the work of Dwork, Ka...

Let F be a totally real field with ring of integers O and p be an odd prime unramified in F . Let p be a prime above p. We prove that a mod p Hilbert modular form associated to F is determined by its restriction to the partial Serre-Tate deformation space Ĝm ⊗ Op (p-rigidity). Let K/F be an imaginary quadratic CM extension such that each prime of F above p splits in K and λ a Hecke character of...