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Single-class genera of positive integral lattices

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  01 August 2013

David Lorch  and
Markus Kirschmer
David Lorch
Affiliation:
Lehrstuhl D für Mathematik,RWTH Aachen University,Templergraben 64,D-52062 Aachen,Germany email david.lorch@math.rwth-aachen.demarkus.kirschmer@math.rwth-aachen.de
Markus Kirschmer
Affiliation:
Lehrstuhl D für Mathematik,RWTH Aachen University,Templergraben 64,D-52062 Aachen,Germany email david.lorch@math.rwth-aachen.demarkus.kirschmer@math.rwth-aachen.de

Abstract

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We give an enumeration of all positive definite primitive $ \mathbb{Z} $-lattices in dimension $n\geq 3$ whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith–Minkowski–Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson.

We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson’s classifications in other dimensions. A list of all single-class primitive $ \mathbb{Z} $-lattices has been compiled and incorporated into the Catalogue of Lattices.

MSC classification

Secondary: 11E12: Quadratic forms over global rings and fields 11E41: Class numbers of quadratic and Hermitian forms
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 172 - 186
Copyright
© The Author(s) 2013 

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