We establish a generalization of the p-adic local monodromy theorem (of André, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called “fake annuli”. The latter correspond loosely to completions of a Laurent polynomial ring with respect to a monomial valuation. The result represents a step towards a higher-dimensiona...

We establish a generalization of the p-adic local monodromy theorem (of André, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called “fake annuli”. The latter correspond loosely to completions of a Laurent polynomial ring with respect to a valuation, which in this paper is restricted to be of monomial form; we defe...

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...

Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp(p) = p−1. When one talks of q-exten...

Throughout this paper Z,Zp,Qp and Cp will be denoted by 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, cf. [7, 8, 9, 10]. Let νp be the normalized exponential valuation of Cp with |p|p = p −νp(p) = p. When one talks of qextension, q is variously considered as an indeterminate, a comple...

Let p be a fixed prime number. Throughout this paper, the symbols Z,Zp,Qp,C, and Cp will denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N ∪ {0}. Let vp be the normalized exponential valuation of Cp with |p|...

Suppose K/Q is a totally real extension of degree d = [K : Q]. Let Qp denote the completion of with respect to the p-adic valuation when p is a rational prime number and let Q,, denote IR when the valuation is the usual absolute value. This latter case is thought of as corresponding to the case where the prime p is 'infinite'. Suppose : K->QP is a (Q-)linear form. We say that is p-ad...

Continued fractions in R have a single definition and algorithms for approximating them are well known. There also exists a well known result which states that √ m, m ∈ Q, always has a periodic continued fraction representation. In Qp, the field of p-adics, however, there are competing and non-equivalent definitions of continued fractions and no single algorithm exists which always produces a p...