Document Type

Article

Department

Mathematics (CMC)

Publication Date

2014

Abstract

We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at s=1 with a real pole of order 2, improving upon a result of Stefan Kühnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less than or equal to N is O(N log N) as N → ∞. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence properties of a zeta-function of similarity classes of such lattices, building on the results of a paper by Glenn Henshaw, Philip Liao, Matthew Prince, Xun Sun, Samuel Whitehead, and the author.

Comments

Please note that the article was first published in the Proceedings of the American Mathematical Society, published by the American Mathematical Society, and is therefore also available at http://www.ams.org/journals/proc/2014-142-02/S0002-9939-2013-11820-4/.

The author was partially supported by a grant from the Simons Foundation (#208969) and by the NSA Young Investigator Grant #1210223 - See more at: http://www.ams.org/journals/proc/2014-142-02/S0002-9939-2013-11820-4/#sthash.uz0vuoUv.dpuf

Rights Information

© 2013 American Mathematical Society

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