Similar Sublattices of Planar Lattices

Baake, Michael ; Scharlau, Rudolf ; Zeiner, Peter

Canadian journal of mathematics, Tome 63 (2011) no. 6, pp. 1220-1237

Résumé

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.

Détail

DOI : 10.4153/CJM-2011-019-9

Mots-clés : 11H06, 11R11, 52C05, 82D25

Bibliographie

[1] [1] Apostol, T. M., Modular functions and Dirichlet series in number theory. Second ed., Graduate Texts in Mathematics, 41, Springer-Verlag, New York, 1990. Google Scholar

[2] [2] Baake, M., Solution of the coincidence problem in dimensions d < 4. In: The mathematics of long-range aperiodic order (Waterloo, ON, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer, Dordrecht, 1997, pp. 9–44. Google Scholar

[3] [3] Baake, M. and Grimm, U., Bravais colourings of planar modules with N-fold symmetry. Z. Krist. 219(2004), no. 2, 72–80. doi:10.1524/zkri.219.2.72.26322 Google Scholar

[4] [4] Baake, M., Heuer, M., Grimm, U., and Zeiner, P., Coincidence rotations of the root lattice A4. European J. Combin. 29(2008), no. 8, 1808–1819. doi:10.1016/j.ejc.2008.01.012 Google Scholar

[5] [5] Baake, M., Heuer, M., and Moody, R. V., Similar sublattices of the root lattice A4. J. Algebra 320(2008), no. 4, 1391–1408. doi:10.1016/j.jalgebra.2008.04.021 Google Scholar

[6] [6] Baake, M. and Moody, R. V., Similarity submodules and semigroups. In: Quasicrystals and discrete geometry (Toronto, ON, 1995). Fields Inst. Monogr., 10, American Mathematical Society, Providence, RI, 1998, pp. 1–13. Google Scholar

[7] [7] Baake, M. and Moody, R. V. Similarity submodules and root systems in four dimensions. Can. J. Math. 51(1999), no. 6, 1258–1276. doi:10.4153/CJM-1999-057-0 Google Scholar

[8] [8] Borewicz, S. I. and Safarevic, I. R., Zahlentheorie. Aus dem Russischen übersetzt von Helmut Koch. Lehrbücher und Monographien aus dem Gebiete der ExaktenWissenschaften, Mathematische Reihe, 32, Birkhäuser Verlag, Basel-Stuttgart, 1966 Google Scholar

[9] [9] Buell, D. A., Binary quadratic forms. Classical theory and modern computations. Springer-Verlag, New York, 1989. Google Scholar

[10] [10] Conway, J. H., Rains, E. M., and Sloane, N. J. A., On the existence of similar sublattices. Can. J. Math. 51(1999), no. 6, 1300–1306. doi:10.4153/CJM-1999-059-5 Google Scholar

[11] [11] Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups. Third ed., Grundlehren der MathematischenWissenschaften, 290, Springer-Verlag, New York, 1999. Google Scholar

[12] [12] Cox, D. A., Primes of the form x2 + ny2. Fermat, class field theory, and complex multiplication. Pure and Applied Mathematics,Wiley-Interscience, New York, 1997. Google Scholar

[13] [13] Glied, S. and Baake, M., Similarity versus coincidence rotations of lattices. Z. Krist. 223(2008), 770–772. Google Scholar

[14] [14] Glied, S., Similarity and coincidence isometries for modules. Canad. Math. Bull., to appear. arXiv:1005.5237. Google Scholar

[15] [15] Hardy, G. H. and Wright, E. M., An introduction to the theory of mumbers. Sixth ed., revised by D. R. Heath-Brown and J. H. Silverman, Oxford University Press, Oxford, 2008. Google Scholar

[16] [16] Huck, C., A note on coincidence isometries of modules in Euclidean space. Z. Krist. 224(2009), 341–344. Google Scholar

[17] [17] Louboutin, S., Minorations (sous l’hypothèse de Riemann généralisée) des nombres de classes des corps quadratique imaginaires. Application. C. R. Acad. Sci. Paris Ser. I Math. 310(1990), no. 12, 795–800. Google Scholar

[18] [18] Marcus, D. A., Number fields. Universitext, Springer-Verlag, New York-Heildelberg, 1977. Google Scholar

[19] [19] Pleasants, P. A. B., Baake, M., and Roth, J., Planar coincidences with N-fold symmetry. J. Math. Phys. 37(1996), no. 2, 1029–1058. doi:10.1063/1.531424 Google Scholar

[20] [20] Scharlau, R., Seminar über komplexe Multiplikation. Parts 1, 2, and 3, available online at http://www.mathematik.uni-dortmund.de/_scharlau/research/ Google Scholar

[21] [21] Serre, J.-P., A course in arithmetic. Graduate Texts in Mathematics, 7, Springer-Verlag, New York-Heidelberg, 1973. Google Scholar

[22] [22] Siegel, C. L., Analytische Zahlentheorie. II. lecture notes, edited by K. F. Kürten and G. Köhler, Univ. Göttingen (1964). Google Scholar

[23] [23] Sloane, N. J. A., The online encyclopedia of integer sequences. http://www.research.att.com/_njas/sequences/. Google Scholar

[24] [24] Tenenbaum, G., Introduction à la théorie analytique et probabiliste des nombres. Second edition. Cours Spécialisés, 1, Société Mathématique de France, Paris, 1995. Google Scholar

[25] [25] Weinberger, P. J., Exponents of class groups of quadratic fields. Acta Arith. 22(1973), 117–124. Google Scholar

[26] [26] Zagier, D. B., Zetafunktionen und quadratische Körper. Hochschultext, Springer-Verlag, Berlin-New York, 1981. Google Scholar

[27] [27] Zeiner, P., Symmetries of coincidence site lattices of cubic lattices. Z. Krist. 220(2005), 915–925. Google Scholar