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An absolute Siegel’s lemma. (English) Zbl 0860.11036
J. Reine Angew. Math. 476, 1-26 (1996); addendum and erratum ibid. 508, 47-51 (1999).
For a number field \(K\), let \(|\cdot |_v\) \((v \in M_K)\) be the absolute values on \(K\) normalised such that if \(v\) lies above the (infinite or finite) prime \(p\) of \(\mathbb{Q}\), then \(|\cdot |_v\) is a continuation of \(|\cdot |_p\) to \(K\). These absolute values satisfy the product formula \(\prod_v |x |^{n_v}_v=1\) for non-zero \(x\in K\), where \(n_v= [K_v: \mathbb{Q}_p]\) and the product is taken over \(v\in M_K\). For \({\mathbf x} =(x_1, \dots, x_n) \in K^n\) define the norms \(|{\mathbf x} |_v= (\sum^n_{i=1} |x_i |^2_v)^{1/2}\) if \(v\) is infinite and \(|{\mathbf x}|_v = \max^n_{i=1} |x_i |_v\) if \(v\) is finite, and the absolute height \(H({\mathbf x})= (\prod_v |{\mathbf x} |^{n_v}_v)^{1/[K: \mathbb{Q}]}\). This height depends only on \({\mathbf x}\), i.e. is otherwise independent of \(K\), and hence can be defined on \(\overline \mathbb{Q}^n\). One defines the height \(H(V)\) of a linear subspace of \(\overline \mathbb{Q}^n\) by taking the height of the exterior product of a basis of \(V\). A reformulation of Siegel’s lemma proved by E. Bombieri and J. Vaaler [Invent. Math. 73, 11-32 (1983; Zbl 0533.10030)] states that if \(V\) is an \(m\)-dimensional linear subspace of \(\overline \mathbb{Q}^n\) and \(K\) is the field of definition of \(V\), then \(V\) has a basis \({\mathbf x}_1, \dots, {\mathbf x}_m\), contained in \(K^n\), such that \(H({\mathbf x}_1) \cdots H({\mathbf x}_m)\leq C(K, m,n) H(V)\), where \(C(K,m,n)\) depends only on \(K,m,n\). In the present paper, the authors prove an “absolute” analogue of this result. More precisely, they show that for every \(\varepsilon >0\), \(V\) has a basis \({\mathbf x}_1, \dots, {\mathbf x}_m\) in \(\overline \mathbb{Q}^n\) such that \[ H({\mathbf x}_1) \cdots H({\mathbf x}_m) < \{2^{m(m-1)} + \varepsilon\} H(V). \] Moreover, the authors show that the constant on the right-hand side can not be improved by anything better than \((m+1)^{-1/2} 2^{m /2}\). As an application, the authors show that between any two given linear subspaces \(W\) and \(V\) of \(\overline \mathbb{Q}^n\) with \(W \subset V\), there is a complete flag of subspaces between \(V\) and \(W\) with small heights with upper bounds depending only on \(m\) and the heights of \(V,W\).
The authors derive their results from a very useful “absolute Minkowski’s theorem”, which is stated in terms of twisted heights. Let \(K_{\mathbf A}\) be the ring of adeles of \(K\) and \(A\in \text{GL} (n,{\mathbf A}_K)\). For \(v \in M_K\), let \(A_v\) be the component of \(A\) at \(v\), so that for every \(v \in M_K\), \(A_v\in \text{GL} (n,K_v)\) and for all but finitely many \(v\), \(A_v \in \text{GL} (n,O_v)\), where \(O_v= \{x\in K_v: |x|_v\leq 1\}\). Define \(|\text{det} A |_{\mathbf A} = (\prod_v |\text{det} A_v |_v^{n_v})^{1/[K: \mathbb{Q}]}\). Define the height of \({\mathbf x} \in K^n\) twisted by \(A\) by \(H_A({\mathbf x}) = (\prod_v |A_v {\mathbf x}|_v^{n_v})^{1/[K: \mathbb{Q}]}\). For every finite extension \(L\) of \(K\), there is a natural embedding \({\mathbf A}_K \hookrightarrow {\mathbf A}_L\), hence \(A\) may be viewed as an element of \(\text{GL} (n,{\mathbf A}_L)\) and so \(H_A({\mathbf x})\) may be defined for \({\mathbf x} \in L^n\); this coincides with the definition on \(K^n\). Thus, \(H_A\) can be continued to a height on \(\overline \mathbb{Q}^n\). Define the successive minima \(\lambda_1, \dots, \lambda_n\) of \(H_A\) with respect to \(\overline \mathbb{Q}^n\) such that for \(i=1, \dots, n\), \(\lambda_i\) is the infimum of all \(\lambda>0\) for which there are \(i\) linearly independent vectors in the set of \({\mathbf x} \in \overline \mathbb{Q}^n\) with \(H_A({\mathbf x}) \leq \lambda\). Then the author shows that \[ |\text{det} A |_{\mathbf A}\leq \prod^n_{i=1} \lambda_i \leq 2^{n(n-1)/2} |\text{det} A|_{\mathbf A}. \] Lastly, the authors prove analogues of the above mentioned results for the algebraic closure of a rational function field in one variable.

MSC:
11H99 Geometry of numbers
11R99 Algebraic number theory: global fields
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