In the present work the notion of the computable (primitive recursive, polynomially time computable) p–adic number is introduced and studied. Basic properties of these numbers and the set of indices representing them are established and it is proved that the above defined fields are p–adically closed. Using the notion of a notation system introduced by Y. Moschovakis an abstract characterizatio...

The amplitudes for the tree-level scattering of the open string tachyons, generalised to the field of p-adic numbers, define the p-adic string theory. There is empirical evidence of its relation to the ordinary string theory in the p → 1 limit. We revisit this limit from a worldsheet perspective and argue that it is naturally thought of as a continuum limit in the sense of the renormalisation g...

Let K be a p-adic field. We explore Igusa’s p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kn. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n = 2 and less than −1 for n = 3 which occur as ...

Let K be a p-adic field of characteristic zero. For polynomials A,B ∈ K[x] we consider decompositions A(x)f(x) +B(x)g(x) = 1 of entire functions f, g on K and try to improve an impossibility result due to A. Boutabaa concerning transcendental f, g. Also, an independent proof of a p-adic diophantic statement due to D. N. Clark is given.

Using a local construction from a previous paper, we exhibit a numerical invariant, the differential Swan conductor, for an isocrystal on a variety over a perfect field of positive characteristic overconvergent along a boundary divisor; this leads to an analogous construction for certain p-adic and l-adic representations of the étale fundamental group of a variety. We then demonstrate some vari...

In this article, I have studied the ultra discrete limit, which is currently studied in soliton theory, from point of view of valuation theory. A quantity obtained after taking the ultra discrete limit should be regarded as non-archimedean valuation, which is related to the p-adic valuation in number theory. The ultra discrete difference-difference equations, whose domain and range are given by...

In [Tei], Teitelbaum formulates a conjecture relating first derivatives of the Mazur– Swinnerton-Dyer p-adic L-functions attached to a modular forms of even weight k ≥ 2 to certain L-invariants arising from Shimura curve parametrisations. This article formulates an analogue of Teitelbaum’s conjecture in which the cyclotomic Zp extension of Q is replaced by the anticyclotomic Zp-extension of an ...