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$ GL_{2}(O_K)$-invariant lattices in the space of binary cubic forms with coefficients in the number field $ K$


Author: Charles A. Osborne
Journal: Proc. Amer. Math. Soc. 142 (2014), 2313-2325
MSC (2010): Primary 11M41, 11R42
DOI: https://doi.org/10.1090/S0002-9939-2014-11978-2
Published electronically: March 19, 2014
MathSciNet review: 3195756
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Abstract: In 2008, Ohno, Taniguchi and Wakatsuki obtained a classification of all $ GL_{2}(\mathbb{Z})$-invariant lattices in the space of binary cubic forms with coefficients in $ \mathbb{Q}$. In this paper, we aim to generalize their result by replacing the rational field with an arbitrary algebraic number field, $ K$. We conclude the paper by connecting the lattices described in our main result to a zeta function developed by Datskovsky and Wright, which yields a functional equation for certain Dirichlet series attached to the lattices.


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Additional Information

Charles A. Osborne
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

DOI: https://doi.org/10.1090/S0002-9939-2014-11978-2
Received by editor(s): May 23, 2011
Received by editor(s) in revised form: August 2, 2012
Published electronically: March 19, 2014
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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