Covolumes, units, regulator: conjectures by D. Bertrand and F. Rodriguez-Villegas
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 \(c_1 > 1\). The examples providing evidence for the validity of Conjecture 1.4 are of two kinds. On the one hand, the work of \textit{E. Friedman} and \textit{N.-P. Skoruppa}, [Invent. Math. 135, No. 1, 115--144 (1999; Zbl 0945.11022)] shows that \(V(K,r_K)\), which equals the regulator of \(K\), is bounded from below exponentially in terms of \(r_K\). In fact, similar estimates are known for relative regulators of number fields, and these are used by the authors to show in Theorem 1.7 of the paper under review that \(\mathrm{Vol}_K(E) \geq 1.1^d\), for every number field \(K\) and every subgroup \(E \subseteq \mathcal{O}_K^\times\), provided that \(E\) contains the subgroup of relative units \(E_{K/F} := \{ \alpha \in \mathcal{O}_K^\times \ \colon \ \mathrm{N}_{K/F}(\alpha) \in F^\times_\text{tors} \}\), for some subfield \(F \subseteq K\) such that \([K \colon F]\) is sufficiently big. On the other hand, work of \textit{M. Pohst}, [Abh. Math. Semin. Univ. Hamb. 47, 95--106 (1978; Zbl 0381.12006)] and \textit{A. Costa} and \textit{E. Friedman}, [J. Number Theory 37, No. 3, 288--297 (1991; Zbl 0718.11060)] shows that the strong version of Conjecture 1.4 holds for totally real fields. In fact, Proposition 3.3 of the paper under review generalises this approach, and shows that the weak version of Conjecture 1.3 is valid as soon as \(K\) belongs to a class of number fields for which the quantity \(u_K := \inf\{ h(\varepsilon) \mid \varepsilon \in \mathcal{O}_K^\times \setminus (\mathcal{O}_K^\times)_\text{tors} \}\) is uniformly bounded away from zero. Such classes of number fields include totally real number fields and CM fields, using a result of \textit{A. Schinzel}, [Acta Arith. 24, 385--399 (1973; Zbl 0275.12004)], and totally \(p\)-adic fields for a fixed prime number \(p\), after a result of \textit{L. Pottmeyer}, [Int. J. Number Theory 14, No. 10, 2687--2697 (2018; Zbl 1441.11174)]. The authors remark that the case \(m = 2\) of Conjecture 1.4 gives an analogue of the conjecture of \textit{A. Schinzel} and \textit{H. Zassenhaus}, [Mich. Math. J. 12, 81--85 (1965; Zbl 0128.03402)], which has recently been proved by [\textit{V. Dimitrov}, ``A proof of the Schinzel-Zassenhaus conjecture on polynomials'', Preprint, \url{arXiv:1912.12545}]. On the other hand, it seems that the validity of Conjecture 1.4 does not imply an analogue of Lehmer's problem, which is due to the fact that Conjecture 1.4 is formulated using the \(L^2\)-norm. Nevertheless, the authors formulate Conjecture 1.10, which is known to be a generalization of Lehmer's problem, and asserts that there exists a function \(c \colon \mathbb{N} \to \mathbb{R}_{> 0}\) such that \(\deg(E) \cdot h(E) \geq c(\mathrm{rk}(E))\) for every subgroup \(E \subseteq \mathcal{O}^\times_{\overline{\mathbb{Q}}}\) which is free and of finite rank. Here, \(\deg(E) := [\mathbb{Q}(E) \colon \mathbb{Q}]\) denotes the degree of the number field generated by the elements of \(E\), and \(h(E) := \mathrm{Vol}_{1,K}(E)/[K \colon \mathbb{Q}]^{\mathrm{rk}(E)}\) denotes the height of \(E\), where \(K\) is any number field containing \(\mathbb{Q}(E)\). To conclude, the paper under review is a good introduction to the mixture of methods coming from analysis and geometry of numbers which are used to study covolumes of subgroups of the logarithmic unit lattice associated to a number field, and provides also a good reference for the newest conjectures in the field.
scientific article; zbMATH DE number 7341602
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