Related papers: An Infinitesimal $p$-adic Multiplicative Manin-Mum…

We prove $p$-adic versions of a classical result in arithmetic geometry stating that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of…

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…

We investigate generalizations along the lines of the Mordell--Lang conjecture of the author's $p$-adic formal Manin--Mumford results for $n$-dimensional $p$-divisible formal groups $\mathcal{F}$. In particular, given a finitely generated…

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions.…

The finite n-th polylogarithm li_n(z) in Z/p[z] is defined as the sum on k from 1 to p-1 of z^k/k^n. We state and prove the following theorem. Let Li_k:C_p to C_p be the p-adic polylogarithms defined by Coleman. Then a certain linear…

We explore systems of polynomial equations where we seek complex solutions with absolute value 1. Geometrically, this amounts to understanding intersections of algebraic varieties with tori -- Cartesian powers of the unit circle. We study…

We prove that for any twist rigid compact $p$-adic analytic group $G$, its twist representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational…

Let $p$ be an idempotent ultrafilter over $\mathbb{N}$. For a positive integer $N$, let ${\cal P}_{\leq N}$ denote the additive group of polynomials $P\in\mathbb{Z}[x]$ with ${\rm deg}\, P\leq N$ and $P(0)=0$. Given a unitary operator $U$…

We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to…

The main objects of study in this paper are those functionals that are analytic in the sense that they annihilate the non-commutative disc algebra. In the classical univariate case, a theorem of F. and M. Riesz implies that such functionals…

We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the…

We establish several results concerning the expected general phenomenon that, given a multiplicative function $f:\mathbb{N}\to\mathbb{C}$, the values of $f(n)$ and $f(n+a)$ are "generally" independent unless $f$ is of a "special" form.…

We analyze certain compositions of rational inner functions in the unit polydisk $\mathbb{D}^{d}$ with polydegree $(n,1)$, $n\in \mathbb{N}^{d-1}$, and isolated singularities in $\mathbb{T}^d$. Provided an irreducibility condition is met,…

In this paper we generalize work of Amice and Lazard from the early (nineteen) sixties. Amice determined the dual of the space of locally Qp-analytic functions on Zp and showed that it is isomorphic to the ring of rigid functions on the…

In this brief note, we consider p-adic unit roots or poles of L-functions of exponential sums defined over finite fields. In particular, we look at the number of unit roots or poles, and a congruence relation on the units. This raises a…

We study a certain truncation of the ring of arithmetical functions with unitary convolution, consisting of functions vanishing on arguments >n. The truncations are artinian monomial quotients of a polynomial ring in finitely many…

We give a short proof of a conjecture of Lubin concerning certain families of $p$-adic power series that commute under composition. We prove that if the family is full (large enough), there exists a Lubin-Tate formal group such that all the…

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet…

This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$\mu_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})/ \mathbb{F}_{q}$. The main…

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ and $\mathcal{S}$ denote the class of functions $f\in\mathcal{A}$ which are univalent ({\it i.e.},…