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Similar Sublattices of Planar Lattices

Published online by Cambridge University Press:  20 November 2018

Michael Baake ,
Rudolf Scharlau  and
Peter Zeiner
Michael Baake
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany email: mbaake@math.uni-bielefeld.de pzeiner@math.uni-bielefeld.de
Rudolf Scharlau
Affiliation:
Fakultät für Mathematik, Universität Dortmund, 44221 Dortmund, Germany email: Rudolf.Scharlau@math.uni-dortmund.de
Peter Zeiner
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany email: mbaake@math.uni-bielefeld.de pzeiner@math.uni-bielefeld.de
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Abstract

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The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

Keywords

11H0611R1152C0582D25

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 6 , 01 December 2011 , pp. 1220 - 1237
Copyright
Copyright © Canadian Mathematical Society 2011

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