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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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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

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