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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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