P-adic Research Articles

Sharpness versus bluntness in affine Weyl groups

In this paper we give an explanation of the bijection between arithmetic and geometric diagrams attached to supercuspidal unipotent representations of a simple p-adic group which is based purely on algebra.

TRANSCENDENTAL RUBAN p-ADIC CONTINUED FRACTIONS

Abstract We establish explicit constructions of Mahler’s p-adic $U_{m}$ -numbers by using Ruban p-adic continued fraction expansions of algebraic irrational p-adic numbers of degree m.

Self-similarity of p-adic groups

We show that a compact open subgroup H of a simple algebraic p-adic group G is self-similar if and only if it is isotropic.

Geometry, Topology and p-Adic Numbers in Geospatial Data Modelling, Management and Processing: Review and Future Approaches

Abstract. Geometry and topology are both central concepts in geospatial data modelling and management. While geometry is the central concept for computational geometry applications, e.g. to intersect surfaces and solids in 3D space, topology is helpful for many application classes starting from city modelling to subsurface modelling and indoor navigation. Furthermore, p-adic numbers are useful for describing hierarchical processes on topological models. In this paper we first shortly review geometry- and topology-based approaches used for geospatial applications. We then describe our approach on turning geometry “upside down” focusing on topology during the whole process of distributed geospatial computing, data modelling, data management, and simulation. Furthermore, the way to use p-adic numbers for the description of hierarchical processes on topological models is shown for the example of simulation and the idea of p-adic analysis in distributed simulations is presented in the context of topological city and building models. The approach opens new insights such as topological relationships, components, and efficiently studying of the approximate behaviour of processes in the built environment using distributed computational systems. Finally, exemplary applications are presented to underline the importance of this new approach.

A local-global principle for similarities over function fields of p-adic curves

Let p∈N be a prime with p≠2, and let K be a p-adic field. Let F be the function field of a curve over K. Let ΩF be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups.

Brownian motion in a vector space over a local field is a scaling limit

For any natural number d, the Vladimirov–Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a d-dimensional vector space V over a local field K. Just as the Laplace operator on L2(Rd) is the infinitesimal generator of Brownian motion with state space Rd, the Vladimirov–Taibleson operator on L2(V) is the infinitesimal generator of real-time Brownian motion with state space V. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted V to be the p-adic numbers.

P -adic reconstruction of rational functions in multiloop amplitudes

Numerical reconstruction techniques are widely employed in the calculation of multiloop scattering amplitudes. In recent years, it has been observed that the rational functions in multiloop calculations greatly simplify under partial fractioning. In this article, we present a technique to reconstruct rational functions directly in partial-fractioned form, by evaluating the functions at special integer points chosen for their properties under a p-adic metric. As an application, we apply this technique to reconstruct the largest rational function in the integration-by-parts reduction of one of the rank-5 integrals appearing in two-loop five-point full-color massless amplitude calculations in quantum chromodynamics. The number of required numerical probes (per prime field) is found to be around 25 times smaller than in conventional techniques, and the obtained result is 130 times smaller. The reconstructed result displays signs of additional structure that could be used to further reduce its size and the number of required probes. Published by the American Physical Society 2024

On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn

In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a p-adic version of this theory started to emerge: there are p-adic Deligne–Lusztig spaces, whose cohomology encodes representation theoretic information for p-adic groups – for instance, it partially realizes the local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic ℓ≠p has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain p-adic Deligne–Lusztig spaces to Vignéras's modular local Langlands correspondence for GLn.

Linear and nonlinear pseudo-differential operators on p-adic fields

Linear and nonlinear pseudo-differential operators on p-adic fields

Kepler sets of second-order linear recurrence sequences over Qp

Let [Formula: see text] be a second-order linear recurrence sequence with constant coefficients over the field of [Formula: see text]-adic numbers [Formula: see text]. We study the set of limit points of the sequence of consecutive ratios [Formula: see text] in [Formula: see text].

A generalization of Kummer theory to Hopf-Galois extensions

We introduce a condition for Hopf-Galois extensions that generalizes the notion of Kummer Galois extension. Namely, an H-Galois extension L/K is H-Kummer if L can be generated by adjoining to K a finite set S of eigenvectors for the action of the Hopf algebra H on L. This extends the classical Kummer condition for the classical Galois structure. With this new perspective, we shall characterize a class of H-Kummer extensions L/K as radical extensions that are linearly disjoint with the n-th cyclotomic extension of K. This result generalizes the description of Kummer Galois extensions as radical extensions of a field containing the n-th roots of the unity. The main tool is the construction of a product Hopf-Galois structure on the compositum of almost classically Galois extensions L1/K, L2/K such that L1∩M2=L2∩M1=K, where Mi is a field such that LiMi=L˜i, the normal closure of Li/K. When L/K is an extension of number or p-adic fields, we shall derive criteria on the freeness of the ring of integers OL over its associated order in an almost classically Galois structure on L/K.

Explicit bounds on torsion of CM abelian varieties over p-adic fields with values in Lubin–Tate extensions

Explicit bounds on torsion of CM abelian varieties over p-adic fields with values in Lubin–Tate extensions

Two-dimensional gauge anomalies and p-adic numbers

We show how methods of number theory can be used to study anomalies in gauge quantum field theories in spacetime dimension two. To wit, the anomaly cancellation conditions for the abelian part of the local anomaly admit solutions if and only if they admit solutions in the reals and in the p-adics for every prime p and we use this to build an algorithm to find all solutions.

Effective field theories on subspaces of the Bruhat-Tits tree

On two subspaces of the Bruhat-Tits tree, effective actions are calculated. The limits of these effective field theories are found to be the same conformal field theory over p-adic numbers when subspaces are taken to the boundary of the tree. Their relations to the p-adic version of AdS/CFT are also discussed.

Asai gamma factors over finite fields

Abstract In this note, we define and study Asai gamma factors over finite fields. We also prove some results about local Asai L-functions over p-adic fields for level zero representations.

P-adic Bessel α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-potentials and some of their applications

In this article, we will study a class of pseudo-differential operators on p-adic numbers, which we will call p-adic Bessel α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-potentials. These operators are denoted and defined in the form (Eϕ,αf)(x)=-Fζ→x-1max{1,|ϕ(||ζ||p)|}-αf^(ζ),x∈Qpn,α∈R,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (\\mathcal {E}_{\\varvec{\\phi },\\alpha }f)(x)=-\\mathcal {F}^{-1}_{\\zeta \\rightarrow x}\\left( \\left[ \\max \\{1,|\\varvec{\\phi }(||\\zeta ||_{p})|\\} \\right] ^{-\\alpha }\\widehat{f}(\\zeta )\\right) , \ ext { } x\\in {\\mathbb {Q}}_{p}^{n}, \\ \\ \\alpha \\in \\mathbb {R}, \\end{aligned}$$\\end{document}where f is a p-adic distribution and max{1,|ϕ(||ζ||p)|}-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left[ \\max \\{1,|\\varvec{\\phi }(||\\zeta ||_{p})|\\}\\right] ^{-\\alpha }$$\\end{document} is the symbol of the operator. We will study some properties of the convolution kernel (denoted as Kα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{\\alpha }$$\\end{document}) of the pseudo-differential operator Eϕ,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {E}_{\\varvec{\\phi },\\alpha }$$\\end{document}, α∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in \\mathbb {R}$$\\end{document}; and demonstrate that the family (Kα)α>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(K_{\\alpha })_{\\alpha >0}$$\\end{document} determines a convolution semigroup on Qpn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_{p}^{n}$$\\end{document}. Furthermore, we will introduce new types of Feller semigroups, and explore new Markov processes and non-homogeneous initial value problems on p-adic numbers.

Cosmic microwave background in a non-Archimedean world

The universe begins its expansion from dimensions below the Planck scale, where the uncertainty principle reigns. If at these dimensions the Archimedean geometry is useless, the p-adic non-Archimedean world creates an environment conducive to phenomenological modeling. The standard description of the cosmological density contrast is made by the Fourier decomposition of plane waves in the field of 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}. The coupling of the primordial perturbations is done by coupling the phases. Since the phases are the stars of this article, we represent them in the hue-saturation-brightness color space both in R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document} and in 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}. If the primordial perturbations interact weakly with each other, with their growth they become important enough from an energetic point of view to gravitationally interact with each other. We identify the coupling using the phase gradient operator, and extract the coupled points from the spectrum. We use pseudoentropy for the quantitative analysis of the information generated by the phases.

A Cauchy Harish-Chandra integral for a dual pair over a p-adic field, the definition and a conjecture

For a real irreducible dual pair there is an integral kernel operator which maps the distribution character of an irreducible admissible representation of the group with the smaller or equal rank to an invariant eigendistribution on the group with the larger or equal rank.The purpose of this article is to transfer this construction to the p-adic case. We provide the precise definition of the integral kernel operator and formulate a conjecture.

Dynamical systems for arithmetic schemes

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vector rings to attach a new ringed space Wrat(X) to every scheme X. We also define R-valued points Wrat(X)(R) of Wrat(X) for every commutative ring R. For normal schemes X of finite type over specZ, using Wrat(X)(ℂ) we construct infinite dimensional R-dynamical systems whose periodic orbits are related to the closed points of X. Various aspects of these topological dynamical systems are studied. We also explain how certain p-adic points of Wrat(X) for X the spectrum of a p-adic local number ring are related to the points of the Fargues–Fontaine curve.

A generalization of arithmetic derivative to p-adic fields and number fields

The arithmetic derivative is a function from the natural numbers to itself that sends all prime numbers to 1 and satisfies the Leibniz rule. The arithmetic partial derivative with respect to a prime p is the p-th component of the arithmetic derivative. In this paper, we generalize the arithmetic partial derivative to p-adic fields (the local case) and the arithmetic derivative to number fields (the global case). We study the dynamical system of the p-adic valuation of the iterations of the arithmetic partial derivatives. We also prove that for every integer n \geq 0, there are infinitely many elements with exactly n anti-partial derivatives. In the end, we study the p-adic continuity of arithmetic derivatives.