Related papers: Representation of units in cyclotomic function fie…

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order…

We construct a q-deformation of the p-adic regulator of a number field, called the cyclosyntomic regulator, building on the Habiro ring of Garoufalidis-Scholze-Wheeler-Zagier. The key new ingredient in our construction is a refinement of…

Fix two distinct primes $p$ and $\ell$. Let $A$ be an abelian variety over $\mathbf{Q}(\zeta_{\ell})$, the cyclotomic field of $\ell$-th roots of unity. Suppose that $A(\mathbf{Q}(\zeta_{\ell}))[\ell] \neq 0$. We show that there exists a…

The main object of study in this paper is the completion Z[q]^N=\varprojlim_n Z[q]/((1-q)(1-q^2)...(1-q^n)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3-spheres with values in Z[q]^N…

Let $L/K$ be a cyclic extension of number fields, and let $S$ be a finite set of places of $K$ containing the ramified and Archimedean ones. We say that $L/K$ has the $\mathbf{cl}^S$-Hilbert 90 property if, for any generator $\sigma \in…

We give a construction of the genus field for Kummer $\ell^n$-cyclic extensions of rational congruence function fields, where $\ell$ is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field,…

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 give some necessary and sufficient conditions for the possibility to represent a Hermitian operator on an infinite-dimensional Hilbert space (real or complex) in the form $\sum_{i=1}^nQ_iP_i$, where $P_1,\dots,P_n$, $Q_1,\dots,Q_n$ are…

Noncompact forms of the Drinfeld-Jimbo quantum groups U_q(g) with (H_i)* = H_i, (X_i^{+-})* = s_i X_i^{-+} for s_i= +-1 are studied at roots of unity. This covers g = so(n,2p), su(n,p), so*(2l), sp(n,p), sp(l,R), and exceptional cases.…

The Gelfand--Zetlin basis for representations of $U_q(sl(N))$ is improved to fit better the case when $q$ is a root of unity. The usual $q$-deformed representations, as well as the nilpotent, periodic (cyclic), semi-periodic (semi-cyclic)…

Let $\M$ be a semi-finite factor and let $\J(\M)$ be the set of operators $T$ in $\M$ such that $T=ETE$ for some finite projection $E$. In this paper we obtain a representation theorem for unitarily invariant norms on $\J(\M)$ in terms of…

This article gives an alternative proof of the fact that N_{Q(zeta)/Q}(1-zeta)=p where p is an odd prime number and zeta is a primitive p-th root of unity, and uses it to prove that N_{Q(zeta)/Q}(1+zeta-zeta^2)=L(p) the p-th Lucas number.…

In the context of cyclotomic fields, it is still unknown whether there exist Euler systems other than the ones derived from cyclotomic units. Nevertheless, we first give an exposition on how norm-compatible units are generated by any Euler…

From some works of P. Furtw\"angler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for…

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…

Let $p$ be an odd prime, and let $\omega$ be a primitive $p$th root of unity. In this paper, we introduce a metric on the cyclotomic field $K=\mathbb{Q}(\omega)$. We prove that this metric has several remarkable properties, such as…

In this work, the commutator of any two reasonable functions of several pairs of canonical conjugate operators is obtained as a sum of terms of partial derivatives of those functions (equations 9, 10 or 11). When applied to quantum…

Let $p$ be a prime number. Let $C_p$, the cyclic group of order $p$, permute transitively a set of indeterminates $\{ x_1,\ldots ,x_p \}$. We prove that the invariant field $\mathbb{Q}(x_1,\ldots ,x_p)^{C_p}$ is rational over $\mathbb{Q}$…

A standard formula (1) leads to a proof of HT90, but requires proving the existence of $\theta$ such that $\alpha\ne 0$, so that $\beta=\alpha/\sigma(\alpha)$. We instead impose the condition (M), that taking $\theta=1$ makes $\alpha=0$.…

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…