On orthogonal hypergeometric groups of degree five

Bajpai, Jitendra; Singh, Sandip

doi:10.1090/tran/7677

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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