Stability of ideal lattices from quadratic number fields
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We study semi-stable ideal lattices coming from quadratic number fields. We prove that all ideal lattices of trace type from rings of integers of imaginary quadratic number fields are semi-stable. For real quadratic fields, we demonstrate infinite families of semi-stable and unstable ideal lattices, 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.
KeywordsSemi-stable lattices Ideal lattices Quadratic number fields
Mathematics Subject Classification11H06 11R11 11E16 11H55
I would like to thank Professor Florian Luca for suggesting the proof of Lemma 2.1, as indicated above. I also thank Professors Gang Yu and David Speyer, whose comments were instrumental to the formulation of Remark 4.1. Finally, I thank the referee for many useful suggestions which improved the quality of the paper.
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