## On orthogonal hypergeometric groups of degree five

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- by Jitendra Bajpai and Sandip Singh PDF
- Trans. Amer. Math. Soc.
**372**(2019), 7541-7572 Request permission

## 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
- 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) - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 7541-7572 - MSC (2010): Primary 22E40; Secondary 32S40, 33C80
- DOI: https://doi.org/10.1090/tran/7677
- MathSciNet review: 4029673