P-adic Research Articles

On p-adic tight wavelet frames

We propose a simple method to constructing step mask and corresponding step wavelet functions that generate tight wavelet frames on the field Qp of p-adic numbers. To construct this mask we use three new ideas. First, we consider only an additive group of the field Qp. Second, to construct the mask we use a special tree. Thirdly, we do not use the principle of unitary extension, we use Pontryagin's principle of duality instead. Using this method, we obtain all step refinable functions.

Trace class operators and states in p-adic quantum mechanics

Within the framework of quantum mechanics over a quadratic extension of the non-Archimedean field of p-adic numbers, we provide a definition of a quantum state relying on a general algebraic approach and on a p-adic model of probability theory. As in the standard complex case, a distinguished set of physical states are related to a notion of trace for a certain class of bounded operators, and in fact, we show that one can define a suitable space of trace class operators in the non-Archimedean setting, as well. The analogies—but also the several (highly non-trivial) differences—with respect to the case of standard quantum mechanics in a complex Hilbert space are analyzed.

Supercuspidal representations in non-defining characteristics

We show that a mod-ℓ-representation of a p-adic group arising from the analogue of Yu's construction is supercuspidal if and only if it arises from a supercuspidal representation of a finite reductive group. This has been previously shown by Henniart and Vigneras under the assumption that the second adjointness holds.

Finitely generated dyadic convex sets

Dyadic rationals are rationals whose denominator is a power of 2. We define dyadic n-dimensional convex sets as the intersections with n-dimensional dyadic space of an n-dimensional real convex set. Such a dyadic convex set is said to be a dyadic n-dimensional polytope if the real convex set is a polytope whose vertices lie in the dyadic space. Dyadic convex sets are described as subreducts (subalgebras of reducts) of certain faithful affine spaces over the ring of dyadic numbers, or equivalently as commutative, entropic and idempotent groupoids under the binary operation of arithmetic mean. The paper contains two main results. First, it is proved that, while all dyadic polytopes are finitely generated, only dyadic simplices are generated by their vertices. This answers a question formulated in an earlier paper. Then, a characterization of finitely generated subgroupoids of dyadic convex sets is provided, and it is shown how to use the characterization to determine the minimal number of generators of certain convex subsets of the dyadic plane.

Geometrically irreducible p-adic local systems are de Rham up to a twist

We prove that any geometrically irreducible Q‾p-local system on a smooth algebraic variety over a p-adic field K becomes de Rham after a twist by a character of the Galois group of K. In particular, for any geometrically irreducible Q‾p-local system on a smooth variety over a number field the associated projective representation of the fundamental group automatically satisfies the assumptions of the relative Fontaine–Mazur conjecture. The proof uses p-adic Simpson and Riemann–Hilbert correspondences of Diao, Lan, Liu, and Zhu and the Sen operator on the decompletions of those developed by Shimizu. Along the way, we observe that a p-adic local system on a smooth geometrically connected algebraic variety over K is Hodge–Tate if its stalk at one closed point is a Hodge–Tate Galois representation. Moreover, we prove a version of the main theorem for local systems with arbitrary geometric monodromy, which allows us to conclude that the Galois action on the proalgebraic completion of π1ét is de Rham.

TOTALLY p-ADIC ALGEBRAIC NUMBERS OF DEGREE 4

We generalize work of Stacy (Open Book Ser. 4 (2020), 387–401). to obtain upper bounds independent of p for the minimal height of a totally p-adic algebraic number of degree 4. We also compute actual values of this minimal height for small primes p.

Generalized Whittaker functions and Jacquet modules

Let $F$ be a non-Archimedean local field and $G$ be (the $F$-points of) a connected reductive group defined over $F$. Fix $U_0$ to be the unipotent radical of a minimal parabolic subgroup $P_0$ of $G$, and $\psi :U_0\rightarrow \mathbb {C}^\times$ be a non-degenerate character of $U_0$. Let $P=MU\supseteq P_0$ be a standard parabolic subgroup of $G$ so that the restriction $\psi _M$ of $\psi$ to $M\cap U_0$ is non-degenerate. We denote by $\mathcal {W}(G,\psi )$ the space of smooth $\psi$-Whittaker functions on $G$ and by $\mathcal {W}_c(G,\psi )$ its $G$-stable subspace consisting of functions with compact support modulo $U_0$. In this situation Bushnell and Henniart identified $\mathcal {W}_c(M,\psi _M^{-1})$ to the Jacquet module of $\mathcal {W}_c(G,\psi ^{-1})$ with respect to $P^-$ (Bushnell and Henniart [Amer. J. Math. 125 (2003), pp. 513–547]). On the other hand Delorme defined a constant term map from $\mathcal {W}(G,\psi )$ to $\mathcal {W}(M,\psi _M)$ which descends to the Jacquet module of $\mathcal {W}(G,\psi )$ with respect to $P$ (Delorme [Trans. Amer. Math. Soc. 362 (2010), pp. 933–955]). We show (as we surprisingly could not find a proof of this statement in the literature) that the descent of Delorme’s constant term map is the dual map of the isomorphism of Bushnell and Henniart, in particular the constant term map is surjective. We also show that the constant term map coincides on admissible submodules of $\mathcal {W}(G,\psi )$ with the inflation of the “germ map” defined by Lapid and Mao [Represent. Theory 13 (2009), pp. 63–81] following earlier works of Casselman and Shalika [Compositio Math. 41 (1980), pp. 207–231]. From these results we derive a simple proof of a slight generalization of a theorem of Delorme and Sakellaridis–Venkatesh ([Ast’erisque 396 (2017), pp. viii+360] for quasi-split $G$) on irreducible discrete series with a generalized Whittaker model to the setting of admissible representations with a central character under the split component of $G$, and similar statements in the cuspidal case (also generalizing a result of Delorme) and in the tempered case. We also show that the germ map of Lapid and Mao is injective, answering one of their questions. Finally using a result of Vignéras [Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 773–801] and recent results of Dat, Helm, Kurinczuk, and Moss [Finiteness for hecke algebras of p-adic groups, arXiv:2203.04929, 2022], we show in the context $\ell$-adic representations that the asymptotic expansion of Lapid and Mao can be chosen to be integral for functions in integral $G$-submodules of $\mathcal {W}(\pi ,\psi )$ of finite length.

The $p$-adic Duffin-Schaeffer conjecture

We prove Haynes' version of the Duffin-Schaeffer conjecture for the $p$-adic numbers. In addition, we prove several results about an associated related but false conjecture, related to $p$-adic approximation in the spirit of Jarník and Lutz.

Zariski density of crystalline points

We show that crystalline points are Zariski dense in the deformation space of a representation of the absolute Galois group of a p-adic field. We also show that these points are dense in the subspace parameterizing deformations with the determinant equal to a fixed crystalline character. Our proof is purely local and works for all p-adic fields and all residual Galois representations.

Restricting admissible representations to fixed-point subgroups

Given a p-adic group and a finite group such that the fixed-point subgroup is reductive, we establish a relationship between constructions of types for G and due to Yu, Kim–Yu and Fintzen, which generalizes the relationship between those of general linear and classical groups originally observed by Stevens. As an application, given an irreducible representation π of G, we explicitly identify a number of the inertial equivalence classes occurring in the representation .

Some further classes of pseudo-differential operators in the p-adic context and their applications.

The purpose of this article is to study new non-Archimedean pseudo-differential operators whose symbols are determined from the behavior of two functions defined on the p-adic numbers. Thanks to the characteristics of our symbols, we can find connections between these operators and new types of non-homogeneous differential equations, Feller semigroups, contraction semigroups and strong Markov processes.

Galois equivariant functions on Galois orbits in large $p$-adic fields

Given a prime number p let \mathbb{C}_p be the topological completion of the algebraic closure of the field of p -adic numbers. Let O(T) be the Galois orbit of a transcendental element T of \mathbb{C}_p with respect to the absolute Galois group. Our aim is to study the class of Galois equivariant functions defined on O(T) with values in \mathbb{C}_p . We show that each function from this class is continuous and we characterize the class of Lipschitz functions, respectively the class of differentiable functions, with respect to a new orthonormal basis. Then we discuss some aspects related to analytic continuation for the functions of this class.

Ergodicity and Periodic Orbits of $$p$$-Adic $$(1,2)$$-Rational Dynamical Systems with Two Fixed Points

We consider $$(1,2)$$ -rational functions given on the field of $$p$$ -adic numbers $${\mathbb Q}_p$$ . In general, such a function has four parameters. We study the case when such a function has two fixed points and show that when there are two fixed points then $$(1,2)$$ -rational function is conjugate to a two-parametric $$(1,2)$$ -rational function. Depending on these two parameters we determine type of the fixed points, find Siegel disks and the basin of attraction of the fixed points. Moreover, we classify invariant sets and study ergodicity properties of the function on each invariant set. We describe 2- and 3-periodic orbits of the $$p$$ -adic dynamical systems generated by the two-parametric $$(1,2)$$ -rational functions.

Coherence of augmented Iwasawa algebras

The augmented Iwasawa algebra of a p-adic Lie group is a generalisation of the Iwasawa algebra of a compact p-adic Lie group. We prove that a split-semisimple group over a p-adic field has a coherent augmented Iwasawa algebra if and only if its root system is of rank one. We deduce that the general linear group of degree n has a coherent augmented Iwasawa algebra precisely when n is at most two. We also characterise when certain solvable p-adic Lie groups have a coherent augmented Iwasawa algebra.

On the generalization of the gap principle

Let \(\alpha \) be a real algebraic number of degree \(d \ge 3\) and let \(\beta \in {\mathbb {Q}}(\alpha )\) be irrational. Let \(\mu \) be a real number such that \((d/2) + 1< \mu < d\) and let \(C_0\) be a positive real number. We prove that there exist positive real numbers \(C_1\) and \(C_2\), which depend only on \(\alpha \), \(\beta \), \(\mu \) and \(C_0\), with the following property. If \(x_1/y_1\) and \(x_2/y_2\) are rational numbers in lowest terms such that $$\begin{aligned} H(x_2, y_2) \ge H(x_1, y_1) \ge C_{1} \end{aligned}$$and $$\begin{aligned} \left| \alpha - \frac{x_1}{y_1}\right|< \frac{C_0}{H(x_1, y_1)^\mu }, \quad \left| \beta - \frac{x_2}{y_2}\right| < \frac{C_0}{H(x_2, y_2)^\mu }, \end{aligned}$$then either \(H(x_2, y_2) > C_{2}^{-1} H(x_1, y_1)^{\mu - d/2}\), or there exist integers s, t, u, v, with \(sv - tu \ne ~0\), such that $$\begin{aligned} \beta = \frac{s\alpha + t}{u\alpha + v} \quad \text {and} \quad \frac{x_2}{y_2} = \frac{sx_1 + ty_1}{ux_1 + vy_1}, \end{aligned}$$or both. Here \(H(x, y) = \max (|x|, |y|)\) is the height of x/y. Since \(\mu - d/2\) exceeds 1, our result demonstrates that, unless \(\alpha \) and \(\beta \) are connected by means of a linear fractional transformation with integer coefficients, the heights of \(x_1/y_1\) and \(x_2/y_2\) have to be exponentially far apart from each other. An analogous result is established in the case when \(\alpha \) and \(\beta \) are p-adic algebraic numbers.

Translation generated oblique dual frames on locally compact groups

Due to the redundancy property of frames, the stable decomposition of a vector in the separable Hilbert space allows the flexibility of choosing different types of duals for a frame. For a second countable locally compact group (not necessarily abelian) and a closed abelian subgroup Γ, we study the properties of oblique Γ-translation generated (Γ-TG) duals for a continuous frame in . Two types of oblique Γ-TG duals viz., type-I and type-II are characterized in terms of the Zak transform for the pair . Outside the group setup, first, we discuss such duals for the multiplication generated systems on the measure-theoretic abstraction in using the range function corresponding to the point-wise conditions in , where X is a σ-finite measure space. Our results present a unified theory connecting the discrete problems with a continuous setup. Besides we characterize these duals' uniqueness using the Gramian/dual Gramian operators, which become a discrete frame/Riesz basis for the associated range space. As an application, we illustrate our results for , p-adic numbers and locally compact abelian groups using fiberization map.

Algebraic geometry and p-adic numbers for scattering amplitude ansätze

Scattering amplitudes in perturbative quantum field theory exhibit a rich structure of zeros, poles and branch cuts which are best understood in complexified momentum space. It has been recently shown that by leveraging this information one can significantly simplify both analytical reconstruction and final expressions for the rational coefficients of transcendental functions appearing in phenomenologically-relevant scattering amplitudes. Inspired by these observations, we present a new algorithmic approach to the reconstruction problem based on p-adic numbers and computational algebraic geometry. For the first time, we systematically identify and classify the relevant irreducible surfaces in spinor space with five-point kinematics, and thanks to p-adic numbers – analogous to finite fields, but with a richer structure to their absolute value – we stably perform numerical evaluations close to these singular surfaces, thus completely avoiding the use of floating-point numbers. Then, we use the data thus acquired to build ansätze which respect the vanishing behavior of the numerator polynomials on the irreducible surfaces. These ansätze have fewer free parameters, and therefore reduced numerical sampling requirements. We envisage future applications to novel two-loop amplitudes.

Simultaneous p-adic Diophantine approximation

AbstractThe aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approximation in the p-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on p-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu in 2019, and a Zero-One law for weighted p-adic approximations established in this paper.

Formula omitted]-adic statistical field theory and deep belief networks

In this work we initiate the study of the correspondence between p-adic statistical field theories (SFTs) and neural networks (NNs). In general quantum field theories over a p-adic spacetime can be formulated in a rigorous way. Nowadays these theories are considered just mathematical toy models for understanding the problems of the true theories. In this work we show these theories are deeply connected with the deep belief networks (DBNs). Hinton et al. constructed DBNs by stacking several restricted Boltzmann machines (RBMs). The purpose of this construction is to obtain a network with a hierarchical structure (a deep learning architecture). An RBM corresponds to a certain spin glass, we argue that a DBN should correspond to an ultrametric spin glass. A model of such a system can be easily constructed by using p-adic numbers. In our approach, a p-adic SFT corresponds to a p-adic continuous DBN, and a discretization of this theory corresponds to a p-adic discrete DBN. We show that these last machines are universal approximators. In the p-adic framework, the correspondence between SFTs and NNs is not fully developed. We point out several open problems.

EEG-based spatio-temporal relation signatures for the diagnosis of depression and schizophrenia

The diagnosis of psychiatric disorders is currently based on a clinical and psychiatric examination (intake). Ancillary tests are used minimally or only to exclude other disorders. Here, we demonstrate a novel mathematical approach based on the field of p-adic numbers and using electroencephalograms (EEGs) to identify and differentiate patients with schizophrenia and depression from healthy controls. This novel approach examines spatio-temporal relations of single EEG electrode signals and characterizes the topological structure of these relations in the individual patient. Our results indicate that the relational topological structures, characterized by either the personal universal dendrographic hologram (DH) signature (PUDHS) or personal block DH signature (PBDHS), form a unique range for each group of patients, with impressive correspondence to the clinical condition. This newly developed approach results in an individual patient signature calculated from the spatio-temporal relations of EEG electrodes signals and might help the clinician with a new objective tool for the diagnosis of a multitude of psychiatric disorders.