P-adic Research Articles

Generation of genetic codes with 2-adic codon algebra and adaptive dynamics

This is a brief review on modeling genetic codes with the aid of 2-adic dynamical systems. In this model amino acids are encoded by the attractors of such dynamical systems. Each genetic code is coupled to the special class of 2-adic dynamics. We consider the discrete dynamical systems, These are the iterations of a function F:Z2→Z2, where Z2 is the ring of 2-adic numbers (2-adic tree). A genetic code is characterized by the set of attractors of a function belonging to the code generating functional class. The main mathematical problem is to reduce degeneration of dynamic representation and select the optimal generating function. Here optimality can be treated in many ways. One possibility is to consider the Lipschitz functions playing the crucial role in general theory of iterations. Then we minimize the Lip-constant. The main issue is to find the proper biological interpretation of code-functions. One can speculate that the evolution of the genetic codes can be described in information space of the nucleotide-strings endowed with ultrametric (treelike) geometry. A code-function is a fitness function; the solutions of the genetic code optimization problem are attractors of the code-function. We illustrate this approach by generation of the standard nuclear and (vertebrate) mitochondrial genetics codes.

GENERIC EXPANSIONS OF GEOMETRIC THEORIES

Abstract As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties $NTP_{2}$ , strongness, $NSOP_{1}$ , and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p-adic fields is $NTP_{2}$ , the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with $\operatorname {\mathrm {bdn}}(\text {"}x=x\text {"})=(\aleph _0)_{-}$ .

Notes on unitary theta representations of compact groups

We continue our work on understanding Howe correspondences by using theta representations from p-adic groups to compact groups. We prove some results for unitary theta representations of compact groups with respect to the induction and restriction functors.

Parameters of Hecke algebras for Bernstein components of [formula omitted]-adic groups

Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra H(O,G) whose category of right modules is closely related to Rep(G)s. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations.In this paper we study the q-parameters of the affine Hecke algebras H(O,G). We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups.Lusztig conjectured that the q-parameters are always integral powers of the cardinality of the residue field of F, and that they coincide with the q-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of absolutely simple p-adic groups, and we prove it for most of those.

A p-adic analog of Hasse--Davenport product relation involving ϵ-factors

Abstract In this paper we prove some generalizations of the classical Hasse–Davenport product relation for certain arithmetic factors defined on a p-adic field F, among them one finds the ϵ-factors appearing in Tate’s thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of SL 2 ⁢ ( F ) {\mathrm{SL}_{2}(F)} to Plancherel measures and γ-factors.

BASIC NONARCHIMEDEAN JØRGENSEN THEORY

Abstract We prove a nonarchimedean analogue of Jørgensen’s inequality, and use it to deduce several algebraic convergence results. As an application, we show that every dense subgroup of ${\mathrm {SL}_2}(K)$ , where K is a p-adic field, contains two elements that generate a dense subgroup of ${\mathrm {SL}_2}(K)$ , which is a special case of a result by Breuillard and Gelander [‘On dense free subgroups of Lie groups’, J. Algebra261(2) (2003), 448–467]. We also list several other related results, which are well known to experts, but not easy to locate in the literature; for example, we show that a nonelementary subgroup of ${\mathrm {SL}_2}(K)$ over a nonarchimedean local field K is discrete if and only if each of its two-generator subgroups is discrete.

Quantum mechanics on a p-adic Hilbert space: Foundations and prospects

We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of [Formula: see text]-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, [Formula: see text]-adic string, Class. Quantum Grav. 4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a [Formula: see text]-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a [Formula: see text]-adic Hilbert space — we consider an algebraic definition of physical states in [Formula: see text]-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a [Formula: see text]-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the [Formula: see text]-adic setting, with an affine geometry; therefore, a symmetry transformation of a [Formula: see text]-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a [Formula: see text]-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.

On integral class field theory for varieties over p-adic fields

Let K be a finite extension of the p-adic numbers Qp with ring of integers OK and residue field κ. Let X a regular scheme, proper, flat, and geometrically irreducible over OK of dimension d, and XK its generic fiber. We show, under some assumptions on X, that there is a reciprocity isomorphism of locally compact groups Har2d−1(XK,Z(d))≃π1ab(XK)W from the cohomology theory defined in [10] to an integral model π1ab(XK)W of the abelianized fundamental group π1ab(XK). After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups. The key ingredient is the duality result in [10].

Continued fractions in the field of 𝑝-adic numbers

Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of p p -adic continued fractions, i.e., continued fractions defined over the field of p p -adic numbers Q p \mathbb {Q}_p , which in the last years has recorded a considerable increase of interest and research activity. We start from the very first definitions up to the most recent developments and open problems.

On Approximation by Tight Wavelet Frames on the Field of $$p$$-Adic Numbers

On Approximation by Tight Wavelet Frames on the Field of $$p$$-Adic Numbers

Computing the Weil representation of a superelliptic curve

We study the Weil representation ρ of a curve over a p-adic field with potential reduction of compact type. We show that ρ can be reconstructed from its stable reduction. For superelliptic curves of the form yn=f(x) at primes p whose residue characteristic is prime to the exponent n we make this explicit.

Transcendence on algebraic groups

Abstract In this paper, we give some new results on transcendence on algebraic groups. These results extend some previous ones established on commutative or linear algebraic groups to arbitrary algebraic groups in complex and p-adic fields, respectively.

Exp function for Edwards curves over local fields

We extend the map Exp for elliptic curves in short Weierstrass form over $ \mathbb{C} $ to Edwards curves over local fields. Subsequently, we compute the map Exp for Edwards curves over the local field $ \mathbb{Q}_{p} $ of $ p $-adic numbers.

The development of P-adic numbers theoretically and their use in number theory

Stemming from the need to formalize the divisibility of rational numbers, p-adics provide a unique mathematical perspective rooted in prime number theory. This article provides a comprehensive discussion of p-adic numbers, including their research background, definition, properties, theory, and extensions. This study first elucidates the historical background and significance of p-adic numbers, emphasizing their key role in number theory and its applications. At the heart of this research is a deep dive into the precise definition of p-radical numbers, revealing their unique, often counterintuitive, distance measure. We look at their fundamental characteristics, demonstrate their illogical characteristics, and discuss how they affect transcendental number theory, Diophantine equations, and algebraic number theory. Furthermore, this article explores the theoretical foundations of p-radical numbers and their extensions, emphasizing their integral role in advanced mathematical structures such as p-radical analysis and p-radical geometry. By covering these aspects, this study aims to highlight the lasting impact of p-adic on modern mathematics, reshape our understanding of divisibility, and advance mathematical inquiry into new and uncharted territory.

States, observables and symmetries in p-adic quantum mechanics

In the framework of quantum mechanics constructed over a quadratic extension of the field of p-adic numbers, we consider an algebraic definition of physical states. Next, the corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a p-adic counterpart of the POVMs associated with a quantum system over the complex numbers. Differently from the standard complex setting, the space of all states of a p-adic quantum system has an affine — rather than convex — structure. Thus, a symmetry transformation may be defined, in a natural way, as a map preserving this affine structure. We argue that the group of all symmetry transformations of a p-adic quantum system has a richer structure wrt the case of standard quantum mechanics over the complex numbers.

Bruhat-Tits buildings, representations of p-adic groups and Langlands correspondence

The Bruhat-Tits theory is a key ingredient in the construction of irreducible smooth representations of p-adic reductive groups. We describe generalizations to arbitrary such representations of several results recently obtained in the case of supercuspidal representations, in particular regarding the local Langlands correspondence and the internal structure of the L-packets.We prove that the enhanced L-parameters with semisimple cuspidal support are those which are obtained via the (ordinary) Springer correspondence.Let G be a connected reductive group over a non-archimedean field F of residual characteristic p. In the case where G splits over a tamely ramified extension of F and p does not divide the order of the Weyl group of G, we show that the enhanced L-parameters with semisimple cuspidal support correspond to irreducible smooth representations of G(F) with non-singular supercuspidal support via the local Langlands correspondence constructed by Kaletha, under the assumption that the latter satisfies certain expected properties of functorial nature. As a consequence, we obtain that every compound L-packet of G(F) contains at least one representation with non-singular supercuspidal support.

Components and Exit Times of Brownian Motion in Two or More p-Adic Dimensions

The fundamental solution to a pseudo-differential equation for functions defined on the d-fold product of the p-adic numbers, Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}, induces an analogue of the Wiener process in Qpd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p^d$$\\end{document}. As in the real setting, the components are 1-dimensional p-adic Brownian motions with the same diffusion constant and exponent as the original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.

Hecke algebras for p-adic reductive groups and Local Langlands Correspondences for Bernstein blocks

We study the endomorphism algebras attached to Bernstein components of reductive p-adic groups and construct a local Langlands correspondence with the appropriate set of enhanced L-parameters, using certain “desiderata” properties for the LLC for supercuspidal representations of proper Levi subgroups. We give several applications of our LLC to various reductive groups with Bernstein blocks cuspidally supported on general linear groups.In particular, for Levi subgroups of maximal parabolic subgroups of the split exceptional group G2, we compute the explicit weight functions for the corresponding Hecke algebras, and show that they satisfy a conjecture of Lusztig's. Some results from §4 are used by the same authors to construct a full local Langlands correspondence in [9]. Moreover, we prove a “reduction to depth-zero” result for regular Bernstein blocks (i.e., blocks for which the supercuspidal support of each irreducible representation is regular).

Some Unipotent Arthur Packets for Reductive p-adic Groups

Abstract Let ${\textsf k}$ be a $p$-adic field, and let $\textbf {G}({\textsf k})$ be the ${\textsf k}$-points of a connected reductive group, inner to split. The set of Aubert–Zelevinsky duals of the constituents of a tempered L-packet forms an Arthur packet for $\textbf {G}({\textsf k})$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang–Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets, which we call weak Arthur packets and describe their constituents in terms of their Langlands parameters.

The set of local 𝐴-packets containing a given representation

Abstract In this paper, we give an algorithm to determine all local 𝐴-packets containing a given irreducible representation of a split special odd orthogonal group or a symplectic group over a 𝑝-adic field. Especially, we can determine whether a given irreducible representation is of Arthur type or not.