Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0103241

Kronecker-Weber plus epsilon

Greg W. Anderson
View original source ↗Read source document ↗Export metadata ↓

Abstract

We say that a group is {\em almost abelian} if every commutator is central and squares to the identity. Now let $G$ be the Galois group of the algebraic closure of the field $\QQ$ of rational numbers in the field of complex numbers. Let $G^{\ab+ε}$ be the quotient of $G$ universal for homomorphisms to almost abelian profinite groups and let $\QQ^{\ab+ε}/\QQ$ be the corresponding Galois extension. We prove that $\QQ^{\ab+ε}$ is generated by the roots of unity, the fourth roots of the (rational) prime numbers and the square roots of certain sine-monomials. The inspiration for the paper came from recent studies of algebraic $Γ$-monomials by P.~Das and by S.~Seo. This paper has appeared as Duke Math. J. 114 (2002) 439-475.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Greg W. Anderson. 2001-03-15. Kronecker-Weber plus epsilon. https://arxiv.org/abs/math/0103241

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Arithmetic Upper and Lower Bounds for the Denumerant Function

We use an elementary arithmetic argument, going back to Blom and Fr{ö}berg, to obtain upper and lower bounds for the denumerant function of Sylvester for arbitrary positive integers $a_1,\dots,a_k$. The bounds hold along the multiples of $\gcd(a_1,\dots,a_k)$, share the leading term of the denumerant, and yield its asymptotic behaviour with an explicit error term; for large $n$ and under a mild condition, a polynomial refinement of the lower bound improves on the lower bound of Agnarsson.

math.NT↗
arXiv2026

The Riemann hypothesis

The Weyl symbolic calculus of operators leads to the construction, if one takes for symbol a certain distribution decomposing over the zeros of the Riemann zeta function, of an operator with the following property: the Riemann hypothesis is equivalent to the validity of a collection of estimates involving this operator. Pseudodifferential arithmetic, a novel chapter of pseudodifferential operator theory, makes it possible to make the operator under study fully explicit. This leads to a proof of the conjecture.

math.NT↗
arXiv2026

On the representation functions of certain numeration systems

Let $β>1$ be fixed. We consider the $(\mathbf{b},\mathcal{D})$-numeration system, where the base sequence $\mathbf{b}=(b_k)_{k\geq 0}$ consists of positive real numbers satisfying $\lim_{k\rightarrow \infty}b_{k+1}/b_k=β$, and the digit set $\mathcal{D}\ni 0$ is a finite set of nonnegative real numbers containing at least two elements. Let $r_{\mathbf{b},\mathcal{D}}(λ)$ denote the number of representations of a given $λ\in\mathbb{R}$ in the form $\sum_{k\ge 0}δ_kb_k$ with digits $δ_k\in \mathcal{D}$. We establish upper bounds for $r_{\mathbf{b},\mathcal{D}}(λ)$ as well as asymptotic formulas for its arbitrary moments. Furthermore, we prove that the associated Dirichlet series $ζ_{\mathbf{b},\mathcal{D}}(s):=\sum_{λ>0}r_{\mathbf{b},\mathcal{D}}(λ)λ^{-s}$ admits a meromorphic continuation to the entire complex plane when $b_k=β^{k}$, and to the half-plane $\mathrm{Re}(s)>\log_β\mathcal{D}-γ$ when $b_k=β^{k}+O(β^{(1-γ)k})$ for any fixed $γ\in(0,1]$. Within the regions of continuation, we determine all candidate poles, compute their corresponding residues, and locate the trivial zeros of $ζ_{\mathbf{b},\mathcal{D}}(s)$. As an application, we resolve several problems posed by Chow and Slattery concerning partitions into distinct terms of certain integer sequences.

math.NT↗