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Similar Sublattices of Planar Lattices

  Published:2011-03-08
 Printed: Dec 2011
  • Michael Baake,
    Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
  • Rudolf Scharlau,
    Fakultät für Mathematik, Universität Dortmund, 44221 Dortmund, Germany
  • Peter Zeiner,
    Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
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Abstract

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.
MSC Classifications: 11H06, 11R11, 52C05, 82D25 show english descriptions Lattices and convex bodies [See also 11P21, 52C05, 52C07]
Quadratic extensions
Lattices and convex bodies in $2$ dimensions [See also 11H06, 11H31, 11P21]
Crystals {For crystallographic group theory, see 20H15}
11H06 - Lattices and convex bodies [See also 11P21, 52C05, 52C07]
11R11 - Quadratic extensions
52C05 - Lattices and convex bodies in $2$ dimensions [See also 11H06, 11H31, 11P21]
82D25 - Crystals {For crystallographic group theory, see 20H15}
 

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