Lower bounds for heights on finitely generated groups
Let \(K\) be an algebraic number field and \(G\) a finitely generated subgroup of the multiplicative group \(K^*\). Suppose that the free part of \(G\) has rank \(r\). Let \(h(\alpha)=\prod_{v\in M_K}\max(1,|\alpha|_v)\) the absolute Weil height of \(\alpha\in K\), where \(|\cdot|_v\) \((v\in M_K)\) are normalised absolute values satisfying the product formula. Previously, the author and \textit{W. M. Schmidt} showed that \(G\) (that is, the free part of \(G\)) has a basis \(\alpha_1,...,\alpha_r\) such that if \(\alpha\) is an arbitrary element of \(G\), expressed uniquely as \(\xi\alpha_1^{x_1}\cdots\alpha_r^{x_r}\) with \(\xi\) a root of unity and \(x_1,...,x_r\in\mathbb{Z}\), then \(h(\alpha)\geq\max_{1\leq i\leq r}\{h(\alpha_i^{x_i})\}^{4^{-r^3}}\) [Math. Ann. 296, No. 2, 339-361 (1993; Zbl 0805.11029)]. In this paper, the author improves this to (*) \(h(\alpha)\geq\max_{1\leq i\leq r}\{h(\alpha_i^{x_i})\}^{4^{-r}}\). The proof uses properties of the height such as \(h(\alpha)>1\) if \(\alpha\) is not a root of unity, \(h(\alpha^n)=h(\alpha)^{|n|}\) for \(n\in\mathbb{Z}\), and \(h(\alpha\beta)\leq h(\alpha)h(\beta)\). Pick any basis \(\beta_1,...,\beta_r\) of \(G\), and define \(\omega({\mathbf x})=\log h(\beta_1^{x_1}\cdots\beta_r^{x_r})\) for \({\mathbf x}\in\mathbb{Z}^r\). Then from the above properties of \(h\) it follows that \(\omega\) can be extended to a vector norm (i.e., distance function) on \(\mathbb{R}^r\). Such a vector norm can be approximated by a vector norm on \(\mathbb{R}^r\) of the shape \(\omega(x)=\sqrt{Q(x)}\), where \(Q\) is a positive definite quadratic form on \(\mathbb{R}^r\). The author proves his theorem by taking a basis \({\mathbf a}_1=(a_{11},...,a_{1r}),...,{\mathbf a}_r\) of \(\mathbb{Z}^r\) which is reduced with respect to \(Q\) in the sense of \textit{A. K. Lenstra, H. W. Lenstra jun.} and \textit{L. Lovász} [Math. Ann. 261, 515-534 (1982; Zbl 0488.12001)] and then showing that \(\alpha_i=\beta_1^{a_{i1}}\cdots\beta_r^{a_{ir}}\) \((i=1,...,r)\) is a basis of \(G\) satisfying (*).
scientific article; zbMATH DE number 986874