Stability of ideal lattices from quadratic number fields. (English) Zbl 1325.11061

Fukshansky, Lenny

Ramanujan J. 37, No. 2, 243-256 (2015).

The \(i\)th successive minimum of a lattice \(\Lambda\) of full rank \(n\) in the Euclidean space \(\mathbb{R}^n\) is defined to be the least \(\lambda_i>0\) such that the vectors of length \(\leq \lambda_i\) in \(\Lambda\) span a sublattice of rank \(\geq i\). So, in particular, \(\lambda_1\) is the minimum length of vectors in the lattice, and \(\lambda_1\leq \lambda_2\leq\ldots\leq\lambda_n\). \(\Lambda\) is called well-rounded if its minimal vectors span \(\mathbb{R}^n\), i.e. \(\lambda_1=\lambda_n\), and it is called semi-stable if for any sublattice \(\Omega\subset\Lambda\) we have that if \(m\leq n\) is the rank of \(\Omega\) then \(\det(\Lambda)^{1/n}\leq\det(\Omega)^{1/n}\).
First, the author shows that well-roundedness implies semi-stability in rank \(2\), but that there are infinitely many similarity classes of unstable well-rounded lattices whenever rank \(n\geq 3\). He then turns his attention to ideal lattices coming from ideals in the ring of integers of quadratic number fields. If \(K\) is a quadratic number field with ring of integers \(\mathcal{O}_K\), then any ideal \(I\subset \mathcal{O}_K\) can be embedded in \(\mathbb{R}^2\) by taking the real and imaginary parts of a given embedding of \(K\) into \(\mathbb{C}\) if \(K\) is imaginary quadratic, or by taking the two conjugate embeddings of \(K\) into \(\mathbb{R}\) if \(K\) is real quadratic, respectively. This gives rise to so-called ideal lattices \(\Lambda_K(I)\) of full rank in \(\mathbb{R}^2\).
The first main result of the paper states that if \(K\) is imaginary quadratic, then such ideal lattices are always semi-stable. The second main (and more difficult) result states that if \(K\) is real quadratic, then there are infinitely many semi-stable as well as infinitely many unstable ideal lattices, and these cases can be distinguished in terms of certain easily computable constants attached to the ideals, which also allows to conclude that the semi-stable case appears with positive probability.

Reviewer: Detlev Hoffmann (Dortmund)

MSC:

11H06	Lattices and convex bodies (number-theoretic aspects)
11R11	Quadratic extensions
11E16	General binary quadratic forms
11H55	Quadratic forms (reduction theory, extreme forms, etc.)

Keywords:

lattice; successive minima of lattices; well-rounded lattice; semi-stable lattice; ideal lattice; quadratic number field

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