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Transactions of the American Mathematical Society

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On orthogonal hypergeometric groups of degree five


Authors: Jitendra Bajpai and Sandip Singh
Journal: Trans. Amer. Math. Soc. 372 (2019), 7541-7572
MSC (2010): Primary 22E40; Secondary 32S40, 33C80
DOI: https://doi.org/10.1090/tran/7677
Published electronically: September 12, 2019
MathSciNet review: 4029673
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Abstract:

A computation shows that there are $77$ (up to scalar shifts) possible pairs of integer coefficient polynomials of degree five having roots of unity as their roots and satisfying the conditions of Beukers and Heckman, so that the Zariski closures of the associated monodromy groups are either finite or the orthogonal groups of non-degenerate and non-positive quadratic forms. Following the criterion of Beukers and Heckman, it is easy to check that only $4$ of these pairs correspond to finite monodromy groups, and only $17$ pairs correspond to monodromy groups, for which the Zariski closure has real rank one.

There are $56$ pairs remaining, for which the Zariski closures of the associated monodromy groups have real rank two. It follows from Venkataramana that $11$ of these $56$ pairs correspond to arithmetic monodromy groups, and the arithmeticity of $2$ other cases follows from Singh. In this article, we show that $23$ of the remaining $43$ rank two cases correspond to arithmetic groups.


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Additional Information

Jitendra Bajpai
Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Address at time of publication: Mathematisches Institut, Georg-August Universität Göttingen, Germany
Email: jitendra@math.uni-goettingen.de

Sandip Singh
Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Address at time of publication: Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India
MR Author ID: 1050960
Email: sandip@math.iitb.ac.in

Keywords: Hypergeometric group, monodromy representation, orthogonal group
Received by editor(s): July 30, 2015
Received by editor(s) in revised form: June 9, 2017
Published electronically: September 12, 2019
Additional Notes: The work of the first author is financially supported by ERC Consolidator grant 648329 (GRANT)
The work of the second author is supported in part by the DST-INSPIRE Faculty Fellowship No. DST/INSPIRE/04/2015/000794 and the SEED Grant No. RD/0515-IRCCSH0-035 (IIT Bombay)
Article copyright: © Copyright 2019 American Mathematical Society