Related papers: On a question of Perlis and Stuart regarding arith…

The main aim of this paper is to study an analogue of the generalized divisor function in a number field $\mathbb{K}$, namely, $\sigma_{\mathbb{K},\alpha}(n)$. The Dirichlet series associated to this function is…

We establish precise relations between Euler systems that are respectively associated to a $p$-adic representation $T$ and to its Kummer dual $T^*(1)$. Upon appropriate specialization of this general result, we are able to deduce the…

Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function…

We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also…

For an algebraic number field K such that prime l splits completely in K we define a regulator R(K) that characterize the subgroup of universal norms from the cyclotomic extension of K in the completed group of S-units of K, where S…

Let $\ell>2$ be a positive integer, $\zeta_\ell$ a primitive $\ell$-th root of unity, and $K$ a number field containing $\zeta_\ell+\zeta_\ell^{-1}$ but not $\zeta_\ell$. In a recent paper, Chonoles et. al. study iterated towers of number…

We consider, for real abelian fields K, the Birch--Tate formula linking the tame kernel \#K\_2(Z\_K) to $\zeta$\_K(-1); we compare, for quadratic and cyclic cubic fields with p=2,3, \#K\_2(\BZ\_K)[p^$\infty$] to the order of the torsion…

In this paper we study the problem of the first moment of the Dedekind zeta function of a number field $K$ and improve the error term. As a ready generalization of our proof, we improve the error term in the Piltz divisor problem.

We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit upper bound for the rank of universal quadratic forms over a given number field $K$, under the assumption…

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group…

We consider pro-isomorphic zeta functions of the groups $\Gamma(\mathcal{O}_K)$, where $\Gamma$ is a unipotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these…

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic…

Given a valuation $v$ with quotient field $K$ and a sequence $\mathcal{K} :K_0\subseteq K_1\subseteq\cdots$ of finite extensions of $K$, we construct a weighted tree $\mathcal{T}(v,\mathcal{K})$ encoding information about the ramification…

In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k_{0}, using the values of the zeta function \zeta_{k_{0}} at negative integers as our ``higher…

Let p and $\ell$ be two distinct primes, F a p-adic field and n an integer. We show that any level 0 block of the category of smooth Z $\ell$-valued representations of GL n (F) is equivalent to the unipotent block of an appropriate product…

Let $F$ be a number field. Let $p$ be a prime number. Washington proved the $\ell$-part of the class numbers in cyclotomic $\mathbb{Z}_p$ extension of $F$ is bounded when $F$ is an abelian number field and $\ell\neq p$ is a prime. By class…

A henselian valued field $K$ is called a tame field if its algebraic closure $\tilde{K}$ is a tame extension, that is, the ramification field of the normal extension $\tilde{K}|K$ is algebraically closed. Every algebraically maximal…

We provide the Krull-Remak-Schmidt decomposition of group algebras of the form $k[G]$ where $k$ is a field, which includes fields with prime characteristic, and $G$ a finite abelian group. We achieved this by studying the geometric…

We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex…

Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We…