Let \(H(x,y,z)\) be one of the norms \(\max\{|x|,|y|,|z|\}\) or \(\sqrt{x^2+y^2+z^2}\) on \({\mathbb{R}}^3\). The authors define a Siegel field as a subfield \(K\) of the field \(\overline{\mathbb{Q}}\) of algebraic numbers which satisfies the following property: there exists a constant \(c(2,K)\) such that, for any \((a,b,c)\in K^3\setminus\{(0,0,0)\}\), there exists \((x,y,z)\in K^3\setminus\{(0,0,0)\}\) with \(ax+by+cz=0\) and \(H(x,y,z)\leq c(2,K) H(a,b,c)^{1/2}\). It is equivalent to select one of the two norms, as long as one does not specify the value of \(c(2,K)\). The classical Siegel lemma, which is usually proved by means of Dirichlet’s pigeonhole principle, shows that the field of rational numbers is a Siegel field. This is also a consequence of Minkowski’s first theorem in the geometry of numbers, as was already known by K. Mahler. More generally, any number field is a Siegel field. According to D. Roy and J. L. Thunder [J. Reine Angew. Math. 476, 1–26 (1996; Zbl 0860.11036)] and to S. Zhang [J. Am. Math. Soc. 8, No. 1, 187–221 (1995; Zbl 0861.14018)] the field \(\overline{\mathbb{Q}}\) of algebraic numbers is a Siegel field. One of the goals of the paper under review is to produce further examples and also to give some counterexamples of this property. The definition above involves only homogeneous linear forms in three variables, but this is sufficient for the validity of general versions of Siegel’s lemma.
The authors work on adelic spaces, namely finite dimensional \(K\)-vector spaces with a norm at each place of \(K\). They restrict to what they call the rigid ones, where the norms are defined by an adelic matrix. To such a rigid adelic space \(E\) are attached notions of heights of points \(H_E:E\to {\mathbb{R}}\) and height \(H(E)\) of \(E\). They prove that for any \(n\geq 1\), there exists a constant \(c(n,K)\) such that if \(E\) is a rigid analytic space over \(K\) of dimension \(n\), then there exists a basis \((e_1,\dots,e_n)\) of \(E\) with \[ H_E(e_1)\cdots H_E(e_n)\leq c(n,K) H(E). \] Their study involves the successive minima related to the height, which they denote by \(\Lambda_i(E)\). They introduce also the minima \(\lambda_i^{\mathrm{BV}}(E)\) of Bombieri-Vaaler, the minima \(\lambda_i(E)\) of Thunder, the minima \(Z_i(E)\) of Zhang and the mixed ones like \(\zeta_i^{\mathrm{BV}}(E)\) (Zhang, Bombieri-Vaaler). They compare these minima. They define a Zhang field as a subfield \(K\) of \(\overline{\mathbb{Q}}\) for which \(Z_1(E)\cdots Z_n(E) H(E)^{-1}\) is bounded above by a constant depending only on the dimension \(n\) of \(E\). For \(\overline{\mathbb{Q}}\) such an upper bound is \(\exp(n(H_n-1)/2)\) where \(H_n=1+(1/2)+\cdots+(1/n)\), as shown by Zhang: hence \(\overline{\mathbb{Q}}\) is a Zhang field. The first theorem of this paper is that a Siegel field which is not a number field is a Zhang field. The authors deduce that a field which satisfies the Northcott property as defined in [E. Bombieri and U. Zannier, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 12, No. 1, 5–14 (2001; Zbl 1072.11077)] and is not a number field is not a Siegel field. The authors show that the constant in Zhang’s Theorem above is sharp, even when one restricts to the first minimum : for any \(\epsilon>0\) and any \(n\geq 1\), there exists a rigid analytic space \(E\) over \(\overline{\mathbb{Q}}\) of dimension \(n\) such that \(\Lambda_1(E)H(E)^{-1/n}\geq \exp((H_n-1)/2)-\epsilon\). They give a number of examples and propose several open problems.