Related papers: p-Adic TGD: Mathematical Ideas

This paper is the first one in the series devoted to the calculation of particle mass spectrum in Topological GeometroDynamics. TGD Universe is critical at quantum level and an attractive idea to realize criticality is via conformal…

In this paper, we offer a brief introduction to the $p$-adic numbers and operations in the metric space defined under the $p$-adic norm. Specifically, we provide a clear description of the derivation of the $p$-adic number via the…

The p-adic description of Higgs mechanism in TGD framework provides excellent predictions for elementary particle and hadrons masses ([email protected] 9410058-62). The gauge group of TGD is just the gauge group of the standard model so…

In this short survey we look at a few basic features of p-adic numbers, somewhat with the point of view of a classical analyst. In particular, with p-adic numbers one has arithmetic operations and a norm, just as for real or complex…

This article introduces a new kind of number systems on $p$-adic integers which is inspired by the well-known $3n+1$ conjecture of Lothar Collatz. A $p$-adic system is a piecewise function on $\mathbb{Z}_p$ which has branches for all…

Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…

$p$-adic continued fractions, as an extension of the classical concept of classical continued fractions to the realm of $p$-adic numbers, offering a novel perspective on number representation and approximation. While numerous $p$-adic…

The purpose of this article is to define and study new invariants of topological spaces: the $p$-adic Betti numbers and the $p$-adic torsion. These invariants take values in the $p$-adic numbers and are constructed from a virtual pro-$p$…

The p-adic numbers have found applications in a wide range of diverse fields of research. In some applications the algebraic properties of p-adics enter as an indispensable ingredient of the theory. Another class of applications has to do…

This paper is a sequel to arXiv:2501.14444, in which we shall give proofs of several results stated in arXiv:2501.14444 (Theorems D--L) which, for brevity and clarity, we postponed to this sequel paper. These results were the following: for…

The p-adic formulation of replica symmetry breaking is presented. In this approach ultrametricity is a natural consequence of the basic properties of the p-adic numbers. Many properties can be simply derived in this approach and p-adic…

The p-adic description of Higgs mechanism provides excellent predictions for elementary particle and hadron masses. In this work the construction of p-adic field theory limit of Quantum TGD is considered. Topological condensate defined as a…

This paper is devoted to the problem of ergodicity of $p$-adic dynamical systems. Our aim is to present criteria of ergodicity in terms of coordinate functions corresponding to digits in the canonical expansion of $p$-adic numbers. The…

The canonical dimension is an invariant attached to admissible representations of p-adic reductive groups, which has only received significant attention in the case of mod-p representations. In the case of complex representations, the…

Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and…

We classify all the zeros and non-zero values of a family of hypergeometric series in the $p$-adic setting. These values of hypergeometric series in the $p$-adic setting lead to transformations of hypergeometric series in the $p$-adic…

Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular,…

In this summary of my talk at Strings 2016, I explain how classical dynamics on an infinite tree graph can be dual to a conformal field theory defined over the $p$-adic numbers. An informal introduction to $p$-adic numbers is followed by a…

To every object $X$ of a symmetric tensor category over a field of characteristic $p>0$ we attach $p$-adic integers $\text{Dim}_+(X)$ and $\text{Dim}_-(X)$ whose reduction modulo $p$ is the categorical dimension $\text{dim}(X)$ of $X$,…

We prove the conjectured compatibility of $p$-adic fundamental lines with specializations at motivic points for a wide class of $p$-adic families of $p$-adic Galois representations (for instance, the families which arise from $p$-adic…