Related papers: The $p$-adic integers as final coalgebra

We continue our study of group algebras acting on $L^p$-spaces, particularly of algebras of $p$-pseudofunctions of locally compact groups. We focus on the functoriality properties of these objects. We show that $p$-pseudofunctions are…

Representation of the Cuntz algebra in the space of (complex valued) functions on p-adic disk is introduced. The relation of this representation and the free coherent states is investigated.

We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{\text{adfin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely…

The $p$-adic logarithm appears in many places in number theory. Hence having a good description of the image of the $p$-adic logarithm could be useful, and in particular, to figure out the image of $1 + \mathfrak{m}_K$, where $K$ is an…

We show that the functor of $p$-typical co-Witt vectors on commutative algebras over a perfect field $k$ of characteristic $p$ is defined on, and in fact only depends on, a weaker structure than that of a $k$-algebra. We call this structure…

We define the $p$-adic trace of certain rank-one local systems on the multiplicative group over $p$-adic numbers, using Sekiguchi and Suwa's unification of Kummer and Artin-Schrier-Witt theories. Our main observation is that, for every…

In this paper, using $p$-adic analysis and $p$-adic L-functions, we show how to extend classical congruences (due to Wilson, Gauss, Dirichlet, Jacobi, Wolstenholme, Glaisher, Morley, Lemher and other people) to modulo $p^k$ for any $k>0$.

This paper studies presentations of the Sierpinski gasket as a final coalgebra for functors on several categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses…

Summation of a large class of the functional series, which terms contain factorials, is considered. We first investigated finite partial sums for integer arguments. These sums have the same values in real and all p-adic cases. The…

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

We introduce the algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as the natural extension of the algebraic entropy for endomorphisms of discrete vector spaces. We show that the…

We consider the matrix ${\frak Z}_P=Z_P+Z_P^t$, where the entries of $Z_P$ are the values of the zeta function of the finite poset $P$. We give a combinatorial interpretation of the determinant of ${\frak Z}_P$ and establish a recursive…

For $G$ a symplectic or orthogonal $p$-adic group (not necessarily split), or an inner form of a general linear $p$-adic group, we compute the endomorphism algebras of some induced projective generators \`a la Bernstein of the category of…

We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer…

The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it…

We show that for any coprime integers $\lambda_1 , \ldots , \lambda_k$ and any finite $A \subset \mathbb{Z}$, one has $$|\lambda_1 \cdot A + \ldots + \lambda_k \cdot A| \geq (|\lambda_1| + \ldots + |\lambda_k|)|A|- C,$$ where $C$ only…

We provide a construction of the fixed points of functors which may not be inital algebras or final coalgebras. For an endofunctor F, this fixed point construction may be expressed as a pair of adjoint functors between F-coalgebras and…

In this article we investigate the number of subrings of $\Z^d$ using subring zeta functions and $p$-adic integration.

Let p be a prime number. This paper introduces the Roquette category R_p of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_p, every finite p-group P admits a canonical direct…

We give a reformulation of the Lehmer conjecture about algebraic integers in terms of a simple counting problem modulo p.