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The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields


Authors: Byeong Moon Kim, Ji Young Kim and Poo-Sung Park
Journal: Math. Comp. 79 (2010), 1123-1144
MSC (2000): Primary 11E39; Secondary 11E20, 11E41
DOI: https://doi.org/10.1090/S0025-5718-09-02287-X
Published electronically: July 16, 2009
MathSciNet review: 2600559
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Abstract: We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields $ \mathbb{Q}(\sqrt{-m})$ for all $ m$. For each imaginary quadratic field $ \mathbb{Q}(\sqrt{-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice $ L$ represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then $ L$ is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13. In addition, we determine the minimal rank of universal Hermitian lattices for all imaginary quadratic fields.


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Additional Information

Byeong Moon Kim
Affiliation: Department of Mathematics, Kangnung-Wonju National University, 123 Chibyondong, Gangneung, Gangwon-Do, Korea
Email: kbm@nukw.ac.kr

Ji Young Kim
Affiliation: School of Mathematics, Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul, 130-722, Korea
Email: jykim@kias.re.kr

Poo-Sung Park
Affiliation: Department of Mathematics Education, Kyungnam University, Masan, Kyungnam, 631-701, Korea
Email: pspark@kyungnam.ac.kr

DOI: https://doi.org/10.1090/S0025-5718-09-02287-X
Keywords: Universal Hermitian form
Received by editor(s): March 28, 2008
Received by editor(s) in revised form: April 14, 2009
Published electronically: July 16, 2009
Additional Notes: The first named author was supported by the Korean Council for University Education, grant funded by Korean Government (MOEHRD) for 2006 Domestic Faculty Exchange.
The second and the third named authors were partially supported by KRF(2005-070-c00004).
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.