The explicit formulas of operations, in particular addition and multiplication, of $p $-adic integers are presented. As applications of the results, at first the explicit formulas of operations of Witt vectors with coefficients in $\mathbb{F}_{2}$ are given; then, through solving a problem of Browkin about the transformation between the coefficients of a $p$-adic integer expressed in the ordinary least residue system and the numerically least residue system, similar formulas for Witt vectors with coefficients in $\mathbb{F}_{3}$ are obtained.
We identify the $p$-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic $p$ as the eigenvalues of a product of special values of a certain matrix of $p$-adic series. That matrix is a product $F(Λ^p)^{-1}F(Λ)$, where the entries in the matrix $F(Λ)$ are $A$-hypergeometric series with integral coefficients and $F(Λ)$ is independent of $p$.
In the recent p-adic q-integral on the p-adic integers' rings was constructed >. The purpose of this paper is to give several interesting integral equation for the p-adic q-integerals on the rings of p-adic integers. As an integral equations, we will treat relations between p-adic q-integral's equations and q-Bernoulli and Euler numbers.
We determine the structure modulo p of the de Rham-Witt complex of a smooth scheme X over a discrete valuation ring of mixed characteristic with log-poles along the special fiber Y and show that the sub-sheaf fixed by the Frobenius is isomorphic to the sheaf of p-adic vanishing cycles. We use this together with results of the second author and Madsen to evaluate the $K$-theory with finite coefficients of the quotient field K of the henselian local ring of X at a generic point of Y. The result affirms the Lichtenbaum-Quillen conjecture for the field K.
In [2], I constructed the p-adic q-integral on Zp. In this paper, we consider the properties of the p-adic invariant p-adic q-integral in the ring of p-adic integers at q=-1. Finally we give the some applications of p-adic q-integration at q=-1. These properties are useful and worthwhile to study Euler numbers and polynomials.
Dress and Siebeneicher gave a significant generalization of the construction of Witt vectors, by producing for any profinite group $G$, a ring-valued functor $\mathbf{W}_G$. This paper gives a concrete interpretation of the rings $\mathbf{W}_{\mathbf{Z}_p^2}(k)$ where $k$ is a field of characteristic $p > 0$ in terms of rings of Lipschitz continuous functions on the $p$-adic upper half plane $\mathbf{P}^1(\mathbf{Q}_p)$. As a consequence we show that the Krull dimensions of the rings $\mathbf{W}_{\mathbf{Z}_p^d}(k)$ are infinite for $d \geq 2$ and we show the Teichmüller representatives form a…
By using Cauchy's formula, it is known that Bernoulli numbers and Euler numbers can be represented by the contour integrals \begin{equation*} \begin{aligned} B_n&=\frac{n!}{2πi}\oint \frac{z}{e^z-1}\frac{d z}{z^{n+1}},\label{condefi}\\[4pt] E_n&=\frac{n!}{2πi}\oint \frac{2e^z}{e^{2z}+1}\frac{d z}{z^{n+1}}, \end{aligned} \end{equation*} while the following Witt's formula represents Euler polynomials through the fermionic $p$-adic integrals $$E_n(a)=\int_{\mathbb Z_p}(x+a)^ndμ_{-1}(x).$$ Base on the above Witt's identity and the binomial theorem, we prove some new identities for the Euler polyno…
We express the classical $p$-adic integers $\hat{\mathbb{Z}_p}$, as a metric space, as a final colagebra to a certain endofunctor. We realize the addition and the multiplication on $\hat{\mathbb{Z}_p}$ as the coalgebra maps from $\hat{\mathbb{Z}_p} \times \hat{\mathbb{Z}_p}$.
We derive an inequality on the discrepancy of sequences on the ring of $p$-adic integers $\ZZ_p$ using techniques from Fourier analysis. The inequality is used to obtain an upper bound on the discrepancy of the sequence $α_n = na +b$, where $a$ and $b$ are elements of $\ZZ_p$. This is a $p$-adic analogue of the classical LeVeque inequality on the circle group $\RR/\ZZ$.
This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$μ_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,μ_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic pro-unipotent harmonic actions. In this part IV we define and study $p$-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$μ_{N}$'s involve sequences of positive integers ; in this IV-1, by consid…
In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials {being} strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of $p$-adic integers from polynomials defined over the integers modulo power…
We find upper and lower bounds on the number of rational points that are $ψ$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of $p$-adic weighted simultaneously approximable points intersected with $p$-adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that $τ$-approximate a $p$-adic integer form a set of resonant points …
In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p^{-1}] is "almost" finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p^{-1}] to a finite étale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p-adic Hodge theory. As an illustr…
We give a sufficient and necessary condition for a p-adic integer to have p-th root in the ring of p-adic integers. The same condition holds clearly for residues modulo p^k. We give a proof that Fermat's last theorem is false for p-adic integers and for residues mod p^k.
We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give two examples of applications for systems involving quadratic and linear forms.
The essential aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of p-adic q-invariant distribution on Zp. Moreover, we give some interesting properties concerning this type maximal operator.
We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of $b$-adic integers, $\mathbb Z_{b}, b\in\mathbb N\setminus\{1\}$, by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the `classical' digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of $t,\mathbf T$ and discrepancy.
Let $f(n)=\sum_k \binom nk^{-1}$. In a previous paper, we defined for a p-adic integer x that f(x) is p-definable if lim $f(x_j)$ exists in $Q_p$, where $x_j$ denotes the mod $p^j$ reduction of $x$. We proved that if p is odd, then -1 is the only element of $Z_p-N$ for which f(x) is p-definable. For p=2, we proved that if the 1's in the binary expansion of x are eventually extraordinarily sparse, then f(x) is 2-definable. Here we present some conjectures that f(x) is 2-definable for many more 2-adic integers. We discuss the extent to which we can prove these conjectures.
For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mat…
Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.
The problem of backward dynamics over the ring of p-adic integers is studied. It is shown that Inverse Limit Theory provides the right framework. Backward iterations of a polynomial with p-adic integer coefficients are constructed by solving congruences modulo powers of p, which inturn are solved by Hensel's lifting lemma.
For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $ν_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $ν_{p}(Q(n))$ is shown to be either periodic or unbounded. The first case corresponds to the situation where $Q$ has no roots in the ring of $p$-adic integers. In the periodic situation, the period length is determined.
The ring of Witt vectors $\mathbb{W} R$ over a base ring $R$ is an important tool in algebraic number theory and lies at the foundations of modern $p$-adic Hodge theory. $\mathbb{W} R$ has the interesting property that it constructs a ring of characteristic $0$ out of a ring of characteristic $p > 1$, and it can be used more specifically to construct from a finite field containing $\mathbb{Z}/p\mathbb{Z}$ the corresponding unramified field extension of the $p$-adic numbers $\mathbb{Q}_p$ (which is unique up to isomorphism). We formalize the notion of a Witt vector in the Lean proof assistant, …
Let n be a positive integer and let p be a prime. We calculate the probability that a random monic polynomial of degree n with coefficients in the ring Z_p of p-adic integers splits over Z_p into linear factors.
The classical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction that takes perfect fields k of prime characteristic p > 0 to p-adically complete discrete valuation rings of characteristic 0 with residue field k and are universal in that sense. A. Dress and C. Siebeneicher generalized this construction by producing a functor W_G attached to any profinite group G. The classical Witt vectors are those attached to the p-adic integers. Here we examine the ring structure of W_G(k) for several examples of pro-p groups G and fields k of characte…
We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give `desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as…
We prove a number of p-adic congruences for the coefficients of powers of a multivariate polynomial f(x) with coefficients in a ring R of characteristic zero. If the Hasse--Witt operation is invertible, our congruences yield p-adic limit formulas which conjecturally describe the Gauss--Manin connection and the Frobenius operator on the unit-root crystal attached to f(x). As a second application, we associate with f(x) formal group laws over R. Under certain assumptions these formal group laws are coordinalizations of the Artin--Mazur functors. (This is a final version which we send for a publi…
The $3x+1$ map $T$ is defined on the $2$-adic integers $\mathbb{Z}_2$ by $T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. It is still unproved that under iteration of $T$ the trajectory of any rational $2$-adic integer is eventually cyclic. A $2$-adic integer is rational if and only if its representation with $1$'s and $0$'s is eventually periodic. We prove that the $3x+1$ conjugacy $Φ$ maps aperiodic $v\in\mathbb{Z}_2$ onto aperiodic $2$-adic integers provided that $\underline{\lim}\;(\frac{h}{\ell})_{\ell=1}^{\infty} > \frac{\ln(2)}{\ln(3)}$ where $h$ is the number of $1$'s in the fi…
Let $K$ be a complete discrete valued field of characteristic zero with residue field $k_K$ of characteristic $p > 0$. Let $L/K$ be a finite Galois extension with the Galois group $G$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. In his paper, Hesselholt conjectured that $H^1(G,W(\sO_L))$ is zero, where $\sO_L$ is the ring of integers of $L$ and $W(\sO_L)$ is the Witt ring of $\sO_L$ w.r.t. the prime $p$. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt's conjecture for all Galois extensions.
Let R be a perfect F_p-algebra, equipped with the trivial norm. Let W(R) be the ring of p-typical Witt vectors over R, equipped with the p-adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy between W(R) and the polynomial ring R[T] equipped with the Gauss norm, in which the role of the structure morphism from R to R[T] is played by the Teichmuller map. For instance, we show that the analytic space associated to R is a strong deformation retract of the space associated to W(R). We also show that each fibre forms a tree under the…
Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Assume that $v_p(f(n))\ge s_1$ and $v_p(g(n))\ge s_2$ hold for all integers $n$ for some $s_1, s_2$ fixed non-negative integers. Let $S$ denote the maximum of $v_p(gcd(f(n),g(n)))$ over all integers $n$. In this paper, we establish multiple lower bound for $v_p(r)$. More specifically, we show that $v_p(r)\ge S-\max{s_1,s_2}+ps_1s_2\frac{p-1}{p-p^{-k}}$, where $k=\lfloor \log_p((p-1)\max\{s_1,s_2\}+1)\rfloor -1$.
We study the dynamic structures of the monomial $x^m$ over the ring of $p$-adic integers for every positive integer $m$ and for primes $p=2,3$ and $5$. The dynamic structures are described by investigating minimal decompositions which consist of minimal subsystems and attracting basins.
Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K = Q(zeta) be the p-cyclotomic field and let O_K be the ring of integers of K. Let pi be the prime ideal of K lying over p. An integer B \in O_K is said singular if B^{1/p} not \in K and if B O_K = b^p where b is an ideal of O_K. An integer B \in O_K is said semi-primary if B = beta mod pi^2 where the natural beta is coprime with p. Let sigma be a Q-isomorphism of the field K generating the Galois group Gal(K/Q). When p is irregular, there exists at least one subgroup Gamma of order p of the class grou…
In this note, we introduce and study the Cartier--Witt stack $\mathrm{WCart}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in terms of quasicoherent sheaf theory on $\mathrm{WCart}_X$ in favorable situations.
This is an account of the algebraic geometry of Witt vectors and arithmetic jet spaces. The usual, "p-typical" Witt vectors of p-adic schemes of finite type are already reasonably well understood. The main point here is to generalize this theory in two ways. We allow not just p-typical Witt vectors but those taken with respect to any set of primes in any ring of integers in any global field, for example. This includes the "big" Witt vectors. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar general…
Let $p$ be an odd prime number. In this paper, we study the growth of the Sylow $p$-subgroups of the even $K$-groups of rings of integers in a $p$-adic Lie extension. Our results generalize previous results of Coates and Ji-Qin, where they considered the situation of a cyclotomic $\mathbb{Z}_p$-extension. Our method of proof differs from these previous work. Their proof relies on an explicit description of certain Galois group via Kummer theory afforded by the context of a cyclotomic $\mathbb{Z}_p$-extension, whereas our approach is via considering the Iwasawa cohomology groups with coefficien…
This paper deals with a multiple version of zeta- and L-functions both in the complex case and in the p-adic case: [I] Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite linear combination of MZFs with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special…
Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete system of residues modulo any power of p.
In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that ${pn \choose n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$.
We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto ω(a)$ be the Teichmüller lift of $\mathbb F_q$, and let $χ:\mathbb{Z}\to \mathbb C_p^\times$ be a finite character of $\mathbb Z_p$. The $L$-function associated to the polynomial $f$ and the so-called twisted character $ω^u\times χ$ is denoted by $L_f(ω^u,χ,s)$. We prove that, when the conductor of the character is large enough, the $p$-adic Newton slopes of this $L$-function form arithmetic progressions.
In this paper, we define the p-adic Euler L-functions using the fermionic p-adic integral on Zp. By computing the values of the p-adic Euler L-functions at negative integers, we show that for Dirichlet characters with odd conductor, this definition is quivalent to the previous definition following Kubata-Leopoldt and Washington's approach. We also study the behavior of p-adic Euler L-functions at positive integers. An interesting thing is that most of the results in Section 11.3.3 of Cohen's book [H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240. S…
Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p>0$. Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. Let $\mathbb{W}_n(\cdot)$ denote the ring of $p$-typical Witt vectors of length $n$. Hesselholt conjectured that the pro-abelian group $\{H^1(G,\mathbb{W}_n(\mathcal{O}_L))\}_{n\geq 1}$ is isomorphic to zero. Hogadi and Pisolkar have recently provided a proof of this conjecture. In this paper, we provide an alternative proof o…
For each prime number $p$, the dynamical behavior of the square mapping on the ring $\mathbb{Z}_p$ of $p$-adic integers is studied. For $p=2$, there are only attracting fixed points with their attracting basins. For $p\geq 3$, there are a fixed point $0$ with its attracting basin, finitely many periodic points around which there are countably many minimal components and some balls of radius $1/p$ being attracting basins. All these minimal components are precisely exhibited for different primes $p$.
In this paper, we study groups of automorphisms of algebraic systems over a set of $p$-adic integers with different sets of arithmetic and coordinate-wise logical operations and congruence relations modulo $p^k,$ $k\ge 1.$ The main result of this paper is the description of groups of automorphisms of $p$-adic integers with one or two arithmetic or coordinate-wise logical operations on $p$-adic integers. To describe groups of automorphisms, we use the apparatus of the $p$-adic analysis and $p$-adic dynamical systems. The motive for the study of groups of automorphism of algebraic systems over $…
We define "splitting functions of level l" for any integer l>0. These functions generalize Dwork's splitting functions : they allow us to represent additive characters of order $p^l$. Then we use these functions to obtain a Stickleberger theorem on the $p$-adic valuation of new Gauss sums.
The 3x+1 function T is defined on the positive integers by $T(x) = \frac{3x+1}{2}$ for x odd and $T(x) = \frac{x}{2}$ for x even. The function T has a natural extension to the 2-adic integers, and there is a continuous function $Φ$ which conjugates T to the 2-adic shift map $σ$. Bernstein and Lagarias conjectured that -1 and 1/3 are the only odd fixed points of $Φ$. In this paper we investigate periodicity associated with $Φ$, a property of the map which is a natural extention of solenoidality. We use it to show that there are nontrivial infinite families of 2-adics that are not fixed points o…
