Related papers: On a construction of $C^1(\mathbb{Z}_p)$ functiona…

Let $G$ be a finite $p$-group. We construct a $G$-extension $K/k$ of number fields such that the $p$-adic completion of the unit group of $K$ has a prescribed $\mathbb{Z}_p[G]$-module structure, up to free direct summands.

For any finite cyclic $p$-group $G$, we will show that every $\mathbb{Z}_p$-torsion free finitely generated $\mathbb{Z}_p[G]$-module appears as $\mathcal{O}_K^\times\otimes_{\mathbb{Z}}\mathbb{Z}_p$ up to $\mathbb{Z}_p[G]$-free direct…

We set up a general framework to study Tate cohomology groups of Galois modules along $\mathbb{Z}_p$-extensions of number fields. Under suitable assumptions on the Galois modules, we establish the existence of a five-term exact sequence in…

Let $K$ be a field complete with respect to a nonarchimedean real-valued norm, and let $L/K$ be an algebraic extension. We show that there is a unique norm on $L$ extending the given norm on $K$, with an explicit description. As an…

We study zeta functions enumerating subalgebras or ideals of Lie algebras over finite field of prime order $\mathbb{F}_p$. We first develop a general blueprint method for computing zeta functions of $\mathbb{F}_p$-Lie algebras, and…

We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of…

We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret's result about the ideal class group problem. Another is a…

For every $p > (1 + \sqrt{5})/2$ we construct a uniformly discrete real sequence $\{\lambda_n\}_{n=1}^\infty$ satisfying $|\lambda_n| = n + o(1)$, a function $g \in L^p(\mathbb{R})$, and continuous linear functionals…

Let $k = \mathbb{Q}(\sqrt {-m})$ and $p \geq 3$ split in $k$. We prove new properties of the $\mathbb{Z}_p$-extensions $K/k$, distinct from the cyclotomic one; we do not assume $K/k$ totally ramified, nor the triviality of the $p$-class…

Let $p$ be an irregular prime. Let $K=\Q(\zeta)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$.…

We construct extensions of the field of rational numbers with the Galois group G_2(F_p) by reducing p-adic representations attached to automorphic representations.

Let $K$ be a finite tamely ramified extension of $\Q_p$ and let $L/K$ be a totally ramified $(\Z/p^n\Z)$-extension. Let $\pi_L$ be a uniformizer for $L$, let $\sigma$ be a generator for $\Gal(L/K)$, and let $f(X)$ be an element of $\O_K[X]$…

Let $\mathcal{V}_p(\lambda)$ be the collection of all functions $f$ defined in the unit disc $\ID$ having a simple pole at $z=p$ where $0<p<1$ and analytic in $\ID\setminus\{p\}$ with $f(0)=0=f'(0)-1$ and satisfying the differential…

Let $k_{\infty}$ be a $\Z_p^d$-extension of a global function field $k$ of characteristic $p$. Let $\Cl_{k_{\infty},p}$ be the $p$ completion of the class group of $k_{\infty}$. We prove that the characteristic ideal of the Galois module…

Let $p$ be an odd prime and $F_{\infty,\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group isomorphic to $\mathbb{Z}_p^r\rtimes\mathbb{Z}_p$, $r\geq 1$. Under certain assumptions, we prove an asymptotic formula for the…

If $\mathfrak{p} \subseteq \mathbb{Z}[\zeta]$ is a prime ideal over $p$ in the $(p^d - 1)$th cyclotomic extension of $\mathbb{Z}$, then every element $\alpha$ of the completion $\mathbb{Z}[\zeta]_\mathfrak{p}$ has a unique expansion as a…

Let $p$ be an odd prime and $L/K$ a $p$-adic Lie extension whose Galois group is of the form $\mathbb{Z}_p^{d-1}\rtimes \mathbb{Z}_p$. Under certain assumptions on the ramification of $p$ and the structure of an Iwasawa module associated to…

We consider the class $\Sigma(p)$ of univalent meromorphic functions $f$ on $\ID$ having simple pole at $z=p\in[0,1)$ with residue 1. Let $\Sigma_k(p)$ be the class of functions in $\Sigma(p)$ which have $k$-quasiconformal extension to the…

We give a survey of a couple known constructions of $p$-adic $L$-functions including Iwasawa's construction from classical Stickelberger elements. We then construct "real" Stickelberger elements, i.e., explicit elements in the Galois group…

We introduce a class of rational functions $A:\,\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ which can be considered as a natural extension of the class of Latt\`es maps and establish basic properties of functions from this class.