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On arithmetic lattices in the plane

Authors: Lenny Fukshansky, Pavel Guerzhoy and Florian Luca
Journal: Proc. Amer. Math. Soc. 145 (2017), 1453-1465
MSC (2010): Primary 11H06, 11G50, 11A25, 11G05
Published electronically: October 18, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the $ j$-invariant corresponding to similarity classes of planar lattices.

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  • [1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
  • [2] Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774
  • [3] Bill Casselman, Stability of lattices and the partition of arithmetic quotients, Asian J. Math. 8 (2004), no. 4, 607–637. MR 2127941
  • [4] J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
  • [5] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447
  • [6] Lenny Fukshansky, Revisiting the hexagonal lattice: on optimal lattice circle packing, Elem. Math. 66 (2011), no. 1, 1–9. MR 2763786,
  • [7] Lenny Fukshansky, Stability of ideal lattices from quadratic number fields, Ramanujan J. 37 (2015), no. 2, 243–256. MR 3341688,
  • [8] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
  • [9] Jean-Marie De Koninck and Florian Luca, Analytic number theory, Graduate Studies in Mathematics, vol. 134, American Mathematical Society, Providence, RI, 2012. Exploring the anatomy of integers. MR 2919246
  • [10] Jacques Martinet, Perfect lattices in Euclidean spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 327, Springer-Verlag, Berlin, 2003. MR 1957723
  • [11] M. Ram Murty and Purusottam Rath, Transcendental numbers, Springer, New York, 2014. MR 3134556
  • [12] C. L. Siegel,
    $ \ddot {\mathrm {U}}$ber die Classenzahl quadratischer Zahlenk $ \ddot {\mathrm {o}}$rper,
    Acta Arith., 1:83-86, 1935.
  • [13] T. N. Venkataramana, Classical modular forms, School on Automorphic Forms on 𝐺𝐿(𝑛), ICTP Lect. Notes, vol. 21, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008, pp. 39–74. MR 2508767

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Additional Information

Lenny Fukshansky
Affiliation: Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, California 91711

Pavel Guerzhoy
Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, Hawaii 96822-2273

Florian Luca
Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africa – and – Centro de Ciencias Matemáticas, UNAM, Morelia, México

Keywords: Arithmetic lattice, well-rounded lattice, semi-stable lattice, height, $j$-invariant
Received by editor(s): November 18, 2015
Received by editor(s) in revised form: June 9, 2016
Published electronically: October 18, 2016
Additional Notes: The first author was partially supported by the NSA grant H98230-1510051
The second author was partially supported by a Simons Foundation Collaboration Grant
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society