Stability of ideal lattices from quadratic number fields

2014/02/12 by Lenny Fukshansky, Fukshansky, Lenny
Mathematics · #11E16 #11H06 #11H55 #11R11 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11E16 #msc:11H06 #msc:11H55 #msc:11R11

paper · pdf · doi:10.48550/arxiv.1402.2738

Theorem 1.2 of the previous version was incorrect as stated, it is now removed

arxiv created 2015/08/05 · arxiv updated 2015/08/06

Abstract

We study semi-stable ideal lattices coming from real quadratic number fields. Specifically, we demonstrate infinite families of semi-stable and unstable ideal lattices of trace type, establishing explicit conditions on the canonical basis of an ideal that ensure stability; in particular, our result implies that an ideal lattice of trace type coming from a real quadratic field is semi-stable with positive probability. We also briefly discuss the connection between stability and well-roundedness of Euclidean lattices.

Related

• Canonical basis twists of ideal lattices from real quadratic number fields
• A complete classification of well-rounded real quadratic ideal lattices
• Well-Rounded Twists of Ideal Lattices from Real Quadratic Fields
• Stability of Quadratic Modules
• Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields