A simplified model of tachyon matter in classical and quantum mechanics is constructed. p-Adic path integral quantization of the model is considered. Recent results in using p-adic analysis, as well as perspectives of an adelic generalization, in the investigation of tachyons are briefly discussed. In particular, the perturbative approach in path integral quantization is proposed.

The study of non-abelian fundamental groups renders it plausible that the principle of Birch and Swinnerton-Dyer, whereby non-vanishing of L-values, in some appropriate sense, accounts for the finiteness of integral points, can eventually be extended to hyperbolic curves. Here we will discuss the very simple case of a genus 1 hyperbolic curve X/Q obtained by removing the origin from an elliptic...

In this article, we study integral structures on p-adic Fourier theory by Schneider and Teitelbaum. As an application of our result, we give a certain integral basis of the space of K-locally analytic functions for any finite extension K of Qp, generalizing the basis of Amice of locally analytic functions on Zp. We also use our result to prove congruences of Bernoulli-Hurwitz numbers at supersi...

The purpose of this paper is to define generalized twisted q-Bernoulli numbers by using padic q-integrals. Furthermore, we construct a q-analogue of the p-adic generalized twisted L-functions which interpolate generalized twisted q-Bernoulli numbers. This is the generalization of Kim’s h-extension of p-adic q-L-function which was constructed in [5] and is a partial answer for the open question ...

Let p be a fixed odd prime number. Throughout this paper Z, Q, Zp, Qp and Cp will respectively denote the ring of rational integers, the field of rational numbers, the ring p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp such that |p|p = p−vp(p) = p−1. If q ∈ Cp, we normally as...

It is noted that an efficient algorithm for calculating a p-adic height could have cryptanalytic applications. Elliptic curves and their generalizations are an active research topic with practical applications in cryptography [1], [2], [3]. If E is an elliptic curve defined over a finite field Fp, where p is prime, and if P and Q are points on the curve E such that Q = nP , then the elliptic cu...

Let Fq be a finite field of cardinality q and characteristic p. Let P (x) be any one-variable Laurent polynomial over Fq of degree (d1, d2) respectively and p d1d2. For any fixed s ≥ 1 coprime to p, we prove that the q-adic Newton polygon of the L-functions of exponential sums of P (xs) has a tight lower bound which we call HodgeStickelberger polygon, depending only on the d1, d2, s and the res...

The Littlewood conjecture in Diophantine approximation claims that inf q≥1 q · ‖qα‖ · ‖qβ‖ = 0 holds for all real numbers α and β, where ‖ · ‖ denotes the distance to the nearest integer. Its p-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that inf q≥1 q · ‖qα‖ · |q|p = 0 holds for every real number α and every prime number p, where | · |p denotes the p-adic absolute value ...