P-adic Case Research Articles

Computation of the Cubic Root of a p-adic Number

In this work, we applied the classical numerical method of the secant in the p-adic case to calculate the cubic root of a p-adic number $a\in\mathbb{Q}_{p}^{\ast }$ where $p$ is a prime number, and this through the calculation of the approximate solution of the equation $x^{3}-a=0$. We also determined the rate of convergence of this method and evaluated the number of iterations obtained in each step of the approximation.Computing both the cubic root and other roots of a p-adic number is useful both for their theoretical values as for their theoretical applications in the field of theoretical computer science and cryptography.

The Berry–Keating Hamiltonian and the local Riemann hypothesis

The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have a real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry–Keating Hamiltonian projected onto the subspace of oscillator eigenfunctions of a lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert–Pólya conjecture. The p-adic case is also discussed.

THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION

AbstractThe Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.

Random $p$-adic Riesz products: Continuity, singularity, and dimension

We study precise conditions for mutual absolute continuity and mutual singularity of two random p-adic Riesz products, defined respectively by two sequences of coefficients a k , b k . Our conditions and assertions are specific to the p-adic case. We also calculate explicitly the Hausdorff dimension, and in case the defining coefficients are constant, we have an integral representation of the dimension formula with a rapid convergence rate p -k .

On the asymptotics of Whittaker functions

We study the asymptotics of Whittaker functions on split groups and relate them to the cuspidal exponents of the representation.

Linear fraction P-Adic and adelic dynamical systems

Using an adelic approach we simultaneously consider real and p-adic aspects of dynamical systems whose states are mapped by linear fractional transformations isomorphic to some subgroups of GL(2, ℚ), SL(2, ℚ) and SL(2, ℤ) groups. In particular, we investigate behaviour of these adelic systems when fixed points are rational. It is shown that any of these rational fixed points is p-adic indifferent for all but a finite set of primes. Thus only for finite number of p-adic cases a rational fixed point may be attractive or repelling. Basins of attraction, the Siegel disks and adelic trajectory are examined. It is also shown that real and p-adic norms of any nonzero rational fixed point are connected by adelic product formula.

Extentions between admissible representations of isometry groups of regular trees

This is the continuation of our previous work (Demir, Geom. Dedicata 105, 189–207, 2004). In this paper, we study extensions between admissible representations of the isometry groups of regular trees. We use the results of Demir, (Geom. Dedicata 105, 189–207, 2004) to show, following Schneider and Stuhler (J. Reine. Angew. Math. 436, 19–32, 1993) in the p-adic case, that extensions between admissible representations of G are finite-dimensional.

Elliptic curves and Hilbert’s tenth problem for algebraic function fields over real andp-adic fields

Let k be a field of characteristic zero, V a smooth, positive-dimensional, quasiprojective variety over k, and D a nonempty effective divisor on V. Let K be the function field of V, and A the semilocal ring of D in K. In this paper, we prove the Diophantine undecidability of: (1) A, in all cases; (2) K, when k is (formally) real and V has a real point; (3) K, when k is a subfield of a p-adic field, for some odd prime p. To achieve this, we use Denef's method: from an elliptic curve E over Q, without complex multiplication, one constructs a quadratic twist E' of E over Q(t), which has Mordell-Weil rank one. Most of the paper is devoted to proving (using a theorem of R. Noot) that one can choose f in K, vanishing at D, such that the group E'(K) deduced from the field extension K/Q(f)=Q(t) is equal to E'(Q(t)). Then we mimic the arguments of Denef (for the real case) and of Kim and Roush (for the p-adic case).

On approximation of [formula omitted]-adic numbers by [formula omitted]-adic algebraic numbers

During the last 10 years the classical Khintchine theorem on approximation of real numbers by rational numbers was generalized for integral polynomials of a fixed degree and for approximation by real algebraic numbers. In this paper we prove a complete analogue of the results for the p-adic case.

Généralisation d'un théorème de R. Perez-Marco au cas p-adique

In this Note, we prove, in the p-adic case, a 0–1 law about the linearization of families ( f t : C p m → C p m ) t ∈ Q p under a polynomial perturbation in t. This gives new examples of non linearizable functions in the p-adic case. To cite this article: D. Vieugué, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Some Finiteness Results in the Representation Theory of Isometry Groups of Regular Trees

Let G denote the isometry group of a regular tree of degree ≥3. The notion of congruence subgroup is introduced and finite generation of the congruence Hecke algebras is proven. Let U be congruence subgroup and \(\mathcal{M}\)(G; U) be the category of smooth representations of G generated by their U-fixed vectors. We also show that this subcategory is closed under taking subquotients. All these results are analogues of well-known results in the case of p-adic groups. It is also shown that the category of admissible representation of G is Noetherian in the sense that every subrepresentation of a finitely generated admissible representation is again finitely generated. Since we want to emphesize the similarities between these groups and p-adic groups, we give the same proofs which also work in the p-adic case whenever possible.

Propagator for the free relativistic particle on Archimedean and no Archimedean spaces

We consider dynamics of a free relativists particle at very short distances treating space-time as Archimedean as well as no Archimedean one. Usual action for the relativistic particle is nonlinear. Meanwhile, in the real case, that system may be treated like a system with quadratic (Hamiltonian) constraint. We perform similar procedure in p-adic case, as the simplest example of a no Archimedean space. The existence of the simplest vacuum state is considered and corresponding Green function is calculated. Similarities and differences between obtained results on both spaces are examined and possible physical implications are discussed. .

P -adic multiple zeta values I.

Our main aim in this paper is to give a foundation of the theory of p-adic multiple zeta values. We introduce (one variable) p-adic multiple polylogarithms by Coleman’s p-adic iterated integration theory. We define p-adic multiple zeta values to be special values of p-adic multiple polylogarithms. We consider the (formal) p-adic KZ equation and introduce the p-adic Drinfel’d associator by using certain two fundamental solutions of the p-adic KZ equation. We show that our p-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the p-adic KZ equation and our p-adic multiple zeta values appear as coefficients of the p-adic Drinfel’d associator. We show various properties of p-adic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the p-adic case, via the p-adic KZ equation.

Invariant subspaces of p-adic Hilbert spaces

In this paper we study the invariant subspaces of p-adic Hilbert spaces. We show some fundamental differences between the complex case and the p-adic case.

On Diophantine Approximations of Dependent Quantities in the p-adic Case

In the present paper, we prove an analog of Khinchin's metric theorem in the case of linear Diophantine approximations of plane curves defined over the ring of \(p\)-adic integers by means of (Mahler) normal functions. We also prove some general assertions needed to generalize this result to the case of spaces of higher dimension.

ON ERGODIC BEHAVIOR OF p-ADIC DYNAMICAL SYSTEMS

Monomial mappings, x ↦ xn, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an analogous result for monomial dynamical systems over p-adic numbers. The process is, however, not straightforward. The result will depend on the natural number n. Moreover, in the p-adic case we will not have ergodicity on the unit circle, but on the circles around the point 1.

Extensions of tori in SL(2)

Let SL(2,F) be the metaplectic two-fold cover of SL(2,F), the special linear group in two variables over a local field IF of characteristic 0. The inverse image T of a maximal torus T in SL(2,F) is an abelian extension ofT by ′1, We consider the question of whether this extension is trivial. More generally we find the minimal subgroup A of the cir for ich the extension is split when considered with coefficient see that ‖A‖ = 2,4 or 8 in the p-adic case. We also ARRIVEE explicit splitting function for the cocycle.

A note on p-adic Carlitz's q-Bernoulli numbers

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.

Differential Equations for Fq-Linear Functions

We study certain classes of equations for Fq-linear functions which are the natural function field counterparts of linear ordinary differential equations. It is shown that, in contrast to both classical and p-adic cases, formal power series solutions have positive radii of convergence near a singular point of an equation. Algebraic properties of the ring of Fq-linear differential operators are also studied.

On Diophantine Approximations of Mock Theta Functions of Third Order

The paper gives bounds for the approximation of the values of Ramanujan's Mock Theta functions of third order and more generally of some q-hypergeometric functions by the elements of an algebraic number field. Simultaneous approximations for the values of q-exponential function are also obtained. All the results are given both in the archimedean and p-adic case.