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

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.

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### References:

[1] | André, Y, On nef and semistable Hermitian lattices, and their behaviour under tensor product, Tohoku Math. J. (2), 63, 629-649, (2011) · Zbl 1256.11030 |

[2] | Bayer-Fluckiger, E, Lattices and number fields, Contemp. Math., 241, 69-84, (1999) · Zbl 0951.11016 |

[3] | Bayer-Fluckiger, E.: Ideal lattices. In: A Panorama of Number Theory or the View from Baker’s Garden (Zurich, 1999), pp. 168-184. Cambridge Univ. Press, Cambridge, 2002 · Zbl 1043.11057 |

[4] | Bayer-Fluckiger, E; Nebe, G, On the Euclidean minimum of some real number fields, J. Théor. Nombres Bordeaux, 17, 437-454, (2005) · Zbl 1161.11032 |

[5] | Borek, T, Successive minima and slopes of Hermitian vector bundles over number fields, J. Number Theory, 113, 380-388, (2005) · Zbl 1100.14513 |

[6] | Buell, D.A.: Binary Quadratic Forms. Springer-Verlag, New York (1989) · Zbl 0698.10013 |

[7] | Casselman, B, Stability of lattices and the partition of arithmetic quotients, Asian J. Math., 8, 607-637, (2004) · Zbl 1086.22007 |

[8] | Fukshansky, L; Henshaw, G; Liao, P; Prince, M; Sun, X; Whitehead, S, On well-rounded ideal lattices, II, Int. J. Number Theory, 9, 139-154, (2013) · Zbl 1296.11085 |

[9] | Fukshansky, L; Petersen, K, On ideal well-rounded lattices, Int. J. Number Theory, 8, 189-206, (2012) · Zbl 1292.11077 |

[10] | Grayson, DR, Reduction theory using semistability, Comment. Math. Helv., 59, 600-634, (1984) · Zbl 0564.20027 |

[11] | James, RD, The distribution of integers represented by quadratic forms, Am. J. Math., 60, 737-744, (1938) · JFM 64.0111.04 |

[12] | Kim, Y.: On Semistability of Root Lattices and Perfect Lattices. Preprint, Univ. Illinois (2009) · Zbl 0951.11016 |

[13] | Lang, S.: Algebraic Number Theory. Springer-Verlag, New York (1994) · Zbl 0811.11001 |

[14] | Stuhler, U, Eine bemerkung zur reduktionstheorie quadratischer formen, Arch. Math. (Basel), 27, 604-610, (1976) · Zbl 0338.10024 |

[15] | Weisstein, E.W.: Landau-Ramanujan constant. From MathWorld—a Wolfram web resource. http://mathworld.wolfram.com/Landau-RamanujanConstant.html |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.