p-Adic dynamical systems of Chebyshev polynomials

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ? _p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T _p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ? _p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T _p in the field ? _p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T _n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.     

P-Adic and Non-Archimedean Product Representations

Two algorithms are introduced and shown to lead to a unique product representation for a given p-adic integer A with leading coefficient 1, as a product of terms \(1+{1\over a_n}\) where the a _n are certain rational numbers. The degree of approximation by the N-th partial product of such terms is investigated, and in some explicit cases the products corresponding to particular types of p-adic integers are characterized. In addition, we consider similar representations for elements of arbitrary complete non-archimedean fields with discrete valuations.     

On the representation of integers by p-adic diagonal forms

Let p be an odd prime, let d be a positive integer such that (d,p?1)=1, let r denote the p-adic valuation of d and let m=1+3+3²+…+3^r. It is shown that for every p-adic integer n the equation Σ_i=1^mX_i^d=n has a nontrivial p-adic solution. It is also shown that for all p-adic units a₁, a₂, a₃, a₄ and all p-adic integers n the equation Σ_i=1⁴a_iX_i^p=n has a nontrivial p-adic solution. A corollary to each of these results is that every p-adic integer is a sum of four pth powers of p-adic integers.     

Gauss sums and multinomial coefficients

Consider a Gauss sum for a finite field of characteristic p, where p is an odd prime. When such a sum (or a product of such sums) is a p-adic integer we show how it can be realized as a p-adic limit of a sequence of multinomial coefficients. As an application we generalize some congruences of Hahn and Lee to exhibit p-adic limit formulae, in terms of multinomial coefficients, for certain algebraic integers in imaginary quadratic fields related to the splitting of rational primes. We also give an example illustrating how such congruences arise from a p-integral formal group law attached to the p-adic unit part of a product of Gauss sums.     

Some 3-adic congruences for binomial sums

We prove some 3-adic congruences for binomial sums,which were conjectured by Zhi-Wei Sun.For example,for any integer m≡1(mod 3)and any positive integer n,we have31n n.1Xk=01mk 2k kmin{3(n),3(m.1).1},where 3(n)denotes the 3-adic order of n.In our proofs,we use several auxiliary combinatorial identities and a series converging to 0 over the 3-adic field.     

The 2-adic affine building of type Ã2 and its finite projections

A certain “free” group U is constructed that is generated by three elements of order 3 which pairwise generate a Frobenius group of order 21 and it is shown that U operates regularly on the affine building of type $\text{A}\text{?}{}_2$ over the field of 2-adic numbers. As a result an infinite series of finite rank 3 geometries is obtained whose rank 2 residues are projective planes of order 2, and which possess a regular automorphism group isomorphic to SL₃(p) or SU₃(p) for some prime p.     

On periodic p-adic distributions

In this paper we introduce a notion of periodic p-adic distribution defined on ? _p — the set of p-adic integers. This periodicity depends on a partition of ? _p . For several concrete partitions we describe corresponding periodic p-adic distributions. Moreover, we construct a periodic p-adic measure.     

Two modularity lifting conjectures for families of Siegel modular forms

For a prime p and a positive integer n, using certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus n. Describing L-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from GSp _r × GSp_2m to GSp _r+2m for some positive integers r and m.     

On ergodicity of p-adic dynamical systems for arbitrary prime p

In this paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of 1-Lipschitz p-adic functions that are defined on (and valuated in) the space ? _p of p-adic integers for any prime p. The conditions are stated in terms of coordinate representations of p-adic functions.     

Simultaneous approximation problems of <Emphasis Type="Italic">p</Emphasis>-adic numbers and <Emphasis Type="Italic">p</Emphasis>-adic knapsack cryptosystems - Alice in <Emphasis Type="Italic">p</Emphasis>-adic numberland

In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l _∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.     

Kato’s Euler system and rational points on elliptic curves I: A p-adic Beilinson formula

This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic étale cohomology, this leads to an alternate proof of the main results of [Br2] and [Ge] which is independent of Kato’s explicit reciprocity law.     

On p-adic multiple zeta and log gamma functions

Text

We define p-adic multiple zeta and log gamma functions using multiple Volkenborn integrals, and develop some of their properties. Although our functions are close analogues of classical Barnes multiple zeta and log gamma functions and have many properties similar to them, we find that our p-adic analogues also satisfy reflection functional equations which have no analogues to the complex case. We conclude with a Laurent series expansion of the p-adic multiple log gamma function for (p-adically) large x which agrees exactly with Barnes?s asymptotic expansion for the (complex) multiple log gamma function, with the fortunate exception that the error term vanishes. Indeed, it was the possibility of such an expansion which served as the motivation for our functions, since we can use these expansions computationally to p-adically investigate conjectures of Gross, Kashio, and Yoshida over totally real number fields.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=I9Bv_CycEd8.     

On p-adic quaternionic Eisenstein series

We show that certain p-adic Eisenstein series for quaternionic modular groups of degree 2 become “real” modular forms of level p in almost all cases. To prove this, we introduce a U(p) type operator. We also show that there exists a p-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of p-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).     

On p-adic Diamond–Euler Log Gamma functions

Text

In this paper, using the fermionic p  -adic integral on _Zp${\mathbb{Z}}_p$, we define the corresponding p-adic Log Gamma functions, so-called p-adic Diamond–Euler Log Gamma functions. We then prove several fundamental results for these p-adic Log Gamma functions, including the Laurent series expansion, the distribution formula, the functional equation and the reflection formula. We express the derivative of p-adic Euler L  -functions at s=0$s=0$ and the special values of p-adic Euler L-functions at positive integers as linear combinations of p-adic Diamond–Euler Log Gamma functions. Finally, using the p-adic Diamond–Euler Log Gamma functions, we obtain the formula for the derivative of the p  -adic Hurwitz-type Euler zeta function at s=0$s=0$, then we show that the p-adic Hurwitz-type Euler zeta functions will appear in the studying for a special case of p  -adic analogue of the (S,T)$(S,T)$-version of the abelian rank one Stark conjecture.

Video

For a video summary of this paper, please click here or visit http://youtu.be/DW77g3aPcFU.     

On recent results of ergodic property for p-adic dynamical systems

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc. In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions. In this paper we present a recent summary of results about the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions. The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball. Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.     

Overconvergent modular sheaves and modular forms for GL 2/F

Given a totally real field F and a prime integer p which is unramified in F, we construct p-adic families of overconvergent Hilbert modular forms (of non-necessarily parallel weight) as sections of, so called, overconvergent Hilbert modular sheaves. We prove that the classical Hilbert modular forms of integral weights are overconvergent in our sense. We compare our notion with Katz’s definition of p-adic Hilbert modular forms. For F = ?, we prove that our notion of (families of) overconvergent elliptic modular forms coincides with those of R. Coleman and V. Pilloni.     

Non-archimedean transportation problems and Kantorovich ultra-norms

We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field Q _p of p-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which yield NA versions of Tkachenko-Uspenskij theorem about free abelian topological groups.     

Adjoint selmer groups as Iwasawa modules

We fix a primep. In this paper, starting from a given Galois representation ? having values inp-adic points of a classical groupG, we study the adjoint action of ? on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(?) of ?. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(?))_/L for each number fieldL. We scrutinize the behavior of Sel(Ad(?))_{/E ∞} as an Iwasawa module for a fixed ? _p -extensionE _∞/E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kähler differentials of the universal nearly ordinary deformation ring of ?. WhenG=GL(2), ? is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.     

Families of Siegel modular forms, <Emphasis Type="Italic">L</Emphasis>-functions and modularity lifting conjectures

For a prime p and a positive integer g, by making use of certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus g and associated p-adic L-functions. Describing L-functions attached to Siegel modular forms and their analytic properties from the point of view of motivic L-functions studied by Deligne and Yoshida, we discuss critical values of the L-functions and p-adic interpolation problems. In particular, we formulate a general conjecture on the existence of the modularity lifting from GSp _r × GSp_2m to GSp _r+2m for some positive integers r and m.