— We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a p-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our ...

In recent years, p-adic numbers are widely used in theoretical and mathematical physics cf. 1–8 , such as string theory, statistical mechanics, turbulence theory, quantum mechanics, and so forth. For a prime number p, let Qp be the field of p-adic numbers. It is defined as the completion of the field of rational numbers Q with respect to the non-Archimedean p-adic norm | · |p. This norm is defi...

Given a projective variety X defined over a finite field, the zeta function of divisors counts all irreducible, codimension one subvarieties of X, each measured by their projective degree. For dim(X) ≥ 2, this is a purely p-adic function. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to the all of Cp. When the divisor class group of X has rank one,...

Let p be a fixed odd 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 the algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp with |p|p = p −νp(p) = 1 p . When one talks of q-extension, q is variously considered as an indeter...

Suppose that G is a connected reductive group over a p-adic field F , that K is a hyperspecial maximal compact subgroup of G(F ), and that V is an irreducible representation of K over the algebraic closure of the residue field of F . We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, Kbiequivariant functions f : G(F ) EndV . These Hecke algebras wer...

The local Langlands conjecture for GL(n) over a p-adic field, n < p.

In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork’s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p. After his pioneer p-adic investigation of the Weil conjectures on the zeta function of an algebraic variety over a finite fi...

We formulate a conjectural p-adic analogue of Borel’s theorem relating regulators for higher K-groups of number fields to special values of the corresponding ζ-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of the conj...

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...

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...