Adelic Minkowski’s second theorem over a division algebra. (English) Zbl 1020.11048

Let \(S\) denote a bounded \(o\)-symmetric convex body in \(\mathbb{R}^n\), with volume \(V(S)\). The successive minima \(\lambda_1, \dots, \lambda_n\) of \(S\) with respect to the lattice \(\mathbb{Z}^n\) are defined by: \[ \lambda_i= \inf\{\lambda >o:\lambda S\text{ contains (at least) }i\text{ linearly independent vectors of }\mathbb{Z}^n\} \] (or, more generally, with respect to a lattice \(\Lambda)\). Minkowski’s second fundamental theorem asserts that \[ {2^n\over n!}\leq \lambda_1 \dots\lambda_n V(S)\leq 2^n. \] The theorem may be generalized [see R. B. McFeat, Geometry of numbers in adèle spaces, Dissertationes Math., Warszawa 88 (1971; Zbl 0229.10014) and E. Bombieri and J. Vaaler, Invent. Math. 73, 11-32 (1983; Zbl 0533.10030)], as follows.
Let \(k\) denote an algebraic number field and let \(E=k^L\). For a \(k\)-lattice \(M\) in \(E\) and a bounded, symmetric convex body, \(S\), in \(E\otimes_\mathbb{Q} \mathbb{R}\), the successive minima, \(\lambda_1,\dots,\lambda_L\) of \(S\) with respect to \(M\) may be defined as before and one then obtains a result analogous to Minkowski’s second theorem.
In the present paper, the authors establish a further generalization of Minkowski’s theorem, in which the underlying vector space is defined over a central division algebra, \(D\), of finite dimension over an algebraic number field, \(k\). Let \(E=D^L\) be a left \(D\)-vector space. A subset of \(D\) is an order of \(D\) if it is a sub-ring containing 1 and a \(k\)-lattice. Let \(\Lambda\) denote an order of \(D\), then a \(k\)-lattice of \(E=D^L\) is a \(\Lambda\)-lattice if it is a finitely generated left \(\Lambda\)-module.
For each place, \(v\), of \(k\), \(k_v\) denotes the completion of \(k\) at \(v\). Let \(d=[k:\mathbb{Q}]\) and \(n^2\) the degree of \(D\) over \(k\). Then \(D_v=D\otimes_kk_v\) and \(D_\infty= \prod_{ v \in P_\infty}D_v\), where \(P_\infty\) denotes the set of infinite places; \(D_A= D_\infty \times D_f\). Let \(\Lambda\) denote an order of \(D\) and \(E=D^L\) is called a \(\Lambda\)-lattice if it is a finitely generated left \(\Lambda\)-module. If \(M\) is a \(\Lambda\)-lattice, then for \(v\in P_f\), \(M_v=\Lambda_v \otimes_\Lambda M\).
With the foregoing notation, the authors now define a convex body, \(S\), in \(E_A=(D_A)^L\) by letting \(S_v\) denote a non-empty, open, convex, bounded symmetric subset of \(E_v\) and then \[ \prod_{v\in P_\infty}S_v \times\prod_{v\in P_f}M_v. \] The successive minima \(\lambda_1,\dots,\lambda_L\) of \(S\) may now be defined in a manner similar to that above, and the volume \(V(S)\) of \(S\) is obtained in terms of the local Haar measures of the \(S_v\).
The authors remark that the inequality \[ (\lambda_1\lambda_2 \dots\lambda_L)^{n^2 d}V(S)\leq 2^{n^2dL} \] may be proved as in the earlier generalization and then prove the inequality \[ \left({\bigl \{(n^2)! \pi^{n^2/2} \bigr\}^L \over(n^2L)! \Gamma(n^2 /2+1)^L} \right)^{r_1} \left({\bigl\{ (2n^2)! (2\pi)^{n^2}\bigr\}^L \over(2n^2 L)!\Gamma (n^2+1)^L} \right)^{r_2} \leq(\lambda_1 \lambda_2 \dots \lambda_L)^{n^2 d}V(S)\bigl( \alpha_\infty(D_\infty/ \Lambda)\bigr)^L, \] where \(\alpha_\infty\) is defined in terms of the local Haar measures ‘at infinity’ and where \(r_1\) denotes the number of real places of \(k,r_2\) the number of imaginary places.


11H06 Lattices and convex bodies (number-theoretic aspects)
11R56 Adèle rings and groups
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[1] Bombieri, E., and Vaaler, J.: On Siegel’s Lemma. Invent. Math., 73 , 11-32 (1983). · Zbl 0533.10030
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