Covolumes, units, regulator: conjectures by D. Bertrand and F. Rodriguez-Villegas
type: Item
instance of: scholarly article
title: Covolumes, units, regulator: conjectures by D. Bertrand and F. Rodriguez-Villegas
zbMATH Open document ID: 1470.11183
author: Francesco Amoroso
author: Sinnou David
published in: Annales Mathématiques du Québec
publication date: 2021-04-30T00:00:00Z
review text: The paper under review provides some new evidence towards the conjectures of Bertrand and Rodriguez-Villegas on the covolumes of sub-lattices of the lattice of units of a number field. These conjectures interpolate between some known lower bounds for the regulator of a number field (which are exponential in its degree), and the conjectural lower bounds for the height of an algebraic number, which are the subject of Lehmer's problem. Let us be more precise. For a fixed number field \(K\), one denotes by \(M_{K,\infty}\) its set of Archimedean places, whose cardinality is written as \(\sigma_K\), and by \(\mathcal{L}_K \colon \mathcal{O}_K^\times \hookrightarrow \mathbb{R}^{\sigma_K}\) the logarithmic embedding, sending \(\alpha \to (d_v \log\lvert \alpha \rvert_v)_{v \in M_{K,\infty}}\) where \(d_v := [K_v \colon \mathbb{R}]\). Then, \(\mathcal{O}_K^\times\) is an abelian group of rank \(r_K := \sigma_K - 1\), contained in the hyperplane \(x_1 + \dots + x_{\sigma_K} = 0\) of \(\mathbb{R}^{\sigma_K}\). Now, for every subgroup \(E \subseteq \mathcal{O}_K^\times\), one can define \(\mathrm{Vol}_{p,K}(E)\) to be the covolume of \(\mathcal{L}_K(E)\) inside the real vector space \(\mathbb{R} \cdot \mathcal{L}_K(E) \subseteq \mathbb{R}^{\sigma_K}\), where the latter is endowed with the norm induced by the \(L^p\)-norm on \(\mathbb{R}^{\sigma_K}\). Moreover, one sets \(V(K,m)\) to be the infimum of \(\mathrm{Vol}_{2,K}(E)\) over all the subgroups \(E \subseteq \mathcal{O}_K^\times\) of rank \(m\). Then, the authors of the paper under review unify previous conjectures of \textit{D. Bertrand}, [J. Aust. Math. Soc., Ser. A 62, No. 2, 198--216 (1997; Zbl 0886.11035)] and \textit{F. Rodriguez-Villegas} (see the appendix of the preprint [\url{arXiv:1903.01384}]), to get Conjecture 1.4, which asserts that for every \(m \geq 2\) one has \(V(K,m) \geq c_0 c_1^m\) for some absolute constants \(c_0, c_1 > 0\). In fact, a stronger version of the same conjecture asserts that \(c_1 > 1\).…
reviewed by: Riccardo Pengo
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 7341602
zbMATH Keywords: geometry of numbers
zbMATH Keywords: heights
zbMATH Keywords: Lehmer's problem
zbMATH Keywords: regulator
Wikidata QID: Q123154401
MaRDI profile type: Publication
full work available at URL: s40316-019-00128-z
OpenAlex ID: W2961564895
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DOI: 10.1007/S40316-019-00128-Z
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description: scientific article; zbMATH DE number 7341602