arXiv:1706.08791v1 [] 27 Jun 2017

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE
JITENDRA BAJPAI AND SANDIP SINGH

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 [1], 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 [1], 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 remaining 56 pairs, for which, the Zariski closures of the associated monodromy groups
have real rank two. It follows from Venkataramana [16] that 11 of these 56 pairs correspond to
arithmetic monodromy groups and the arithmeticity of 2 other cases follows from Singh [11]. In
this article, we show that 23 of the remaining 43 rank two cases correspond to arithmetic groups.

1. Introduction
We consider the monodromy action of the fundamental group π1 of P1 (C)\{0, 1, ∞} on the
solution space of the n Fn−1 type hypergeometric differential equation
(1.1)

D(α; β)w = 0

on P1 (C), having regular singularities at the points {0, 1, ∞}, where the differential operator D(α; β)
is defined as the following:
D(α; β) := (θ + β1 − 1) · · · (θ + βn − 1) − z(θ + α1 ) · · · (θ + αn )
d
where θ = z dz
, and α = (α1 , α2 , . . . , αn ), β = (β1 , β2 , . . . , βn ) ∈ Qn .

The monodromy groups of the hypergeometric differential equations (cf. Equation (1.1)) are also
called the hypergeometric groups, which are characterized by the following theorem of Levelt ([6];
cf. [1, Theorem 3.5]):
Theorem 1 (Levelt). If α1 , α2 , . . . , αn , β1 , β2 , . . . , βn ∈ Q such that αj − βk 6∈ Z, for all j, k =
1, 2, . . . , n, then there exists a basis of the solution space of the differential equation (1.1), with
respect to which, the hypergeometric group is a subgroup of GLn (C) generated by the companion
2010 Mathematics Subject Classification. Primary: 22E40; Secondary: 32S40; 33C80.
Key words and phrases. Hypergeometric group, Monodromy representation, Orthogonal group.
1

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JITENDRA BAJPAI AND SANDIP SINGH

matrices A and B of the polynomials
(1.2)

f (X) =

n
Y

2πiαj

(X − e

j=1

)

and

g(X) =

n
Y

(X − e2πiβj )

j=1

respectively, and the monodromy is defined by g∞ 7→ A, g0 7→ B −1 , g1 7→ A−1 B, where g0 , g1 , g∞
are, respectively, the loops around 0, 1, ∞, which generate the π1 of P1 (C)\{0, 1, ∞} modulo the
relation g∞ g1 g0 = 1.
We now denote the hypergeometric group by Γ(f, g) which is a subgroup of GLn (C) generated by
the companion matrices of the polynomials f, g. We consider the cases where the coefficients of f, g
are integers with f (0) = ±1, g(0) = ±1 (for example, one may take f, g as product of cyclotomic
polynomials); in these cases, Γ(f, g) ⊂ GLn (Z). We also assume that f, g form a primitive pair [1,
Definition 5.1] and do not have any common root.
Beukers and Heckman [1, Theorem 6.5] have completely determined the Zariski closures G of
the hypergeometric groups Γ(f, g) which is briefly summarized as follows:
• If n is even and f (0) = g(0) = 1, then the hypergeometric group Γ(f, g) preserves a nondegenerate integral symplectic form Ω on Zn and Γ(f, g) ⊂ SpΩ (Z) is Zariski dense, that is,
G = SpΩ .
(0)
= −1, then Γ(f, g) preserves a non-degenerate integral quadratic
• If Γ(f, g) is infinite and fg(0)
n
form Q on Z and Γ(f, g) ⊂ OQ (Z) is Zariski dense, that is, G = OQ .
• It follows from [1, Corollary 4.7] that Γ(f, g) is finite if and only if either α1 < β1 < α2 <
β2 < · · · < αn < βn or β1 < α1 < β2 < α2 < · · · < βn < αn ; and in this case, we say that
the roots of f and g interlace on the unit circle.
We call a hypergeometric group Γ(f, g) arithmetic, if it is of finite index in G(Z); and thin, if it
has infinite index in G(Z) [9].
The following question of Sarnak [9] has drawn attention of many people: determine the pairs of
polynomials f, g, for which, the associated hypergeometric group Γ(f, g) is arithmetic. There have
been some progress to answer this question.
For the symplectic cases: infinitely many arithmetic Γ(f, g) in Spn (for any even n) are given by
Singh and Venkataramana in [13]; some other examples of arithmetic Γ(f, g) in Sp4 are given by
Singh in [10] and [12]; 7 examples of thin Γ(f, g) in Sp4 are given by Brav and Thomas in [2]; and
in [3], Hofmann and van Straten have determined the index of Γ(f, g) in Sp4 (Z) for some of the
arithmetic cases of [10] and [13].
For the orthogonal cases: when the quadratic form Q has signature (n − 1, 1), Fuchs, Meiri and
Sarnak give 7 (infinite) families (depending on n odd and ≥ 5) of thin Γ(f, g) in [5]; when the
quadratic form Q has signature (p, q) with p, q ≥ 2, 11 (infinite) families of arithmetic Γ(f, g) are
given by Venkataramana in [16]; an example of thin Γ(f, g) in O(2, 2) is given by Fuchs in [4];
and 2 examples of arithmetic Γ(f, g) in O(3, 2) are given by Singh in [11], which deals with the

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

3

14 orthogonal hypergeometric groups of degree five with a maximally unipotent monodromy, and
these cases were inspired by the 14 symplectic hypergeometric groups associated to Calabi-Yau
threefolds [10].
This article resulted from our effort to first write down all possible pairs f, g (up to scalar shifts)
of degree five integer coefficient polynomials, having roots of unity as their roots, forming a primitive
pair and satisfying the condition: f (0) = −1 and g(0) = 1; and then to answer the question to
determine the arithmeticity or thinness of the associated hypergeometric groups Γ(f, g). Note that,
these pairs of the polynomials f, g satisfy the conditions of Beukers and Heckman [1] and hence
the associated hypergeometric group Γ(f, g) is either finite or preserves a non-degenerate integral
quadratic form Q on Z5 , and Γ(f, g) ⊂ OQ (Z) is Zariski dense in the orthogonal group OQ of the
quadratic form Q.
It is clear that there are finitely many pairs of degree five integer coefficient polynomials having
roots of unity as their roots; among these pairs we find that there are 77 pairs (cf. Tables 1, 2,
3, 4, 5, 6, 7) which satisfy the conditions of the last paragraph. Now, we consider the question to
determine the arithmeticity or thinness of the associated orthogonal hypergeometric groups Γ(f, g).
We note that the roots of the polynomials f and g interlace for the 4 pairs f, g of Table 5
and hence the hypergeometric groups associated to these pairs are finite [1, Corollary 4.7]. The
quadratic forms associated to the 17 pairs of Tables 1 and 7 have signature (4, 1), and the thinness
of Γ(f, g) associated to the 7 pairs of Table 1 follows from Fuchs, Meiri, and Sarnak [5]. The
arithmeticity or thinness of Γ(f, g) associated to the other 10 pairs of Table 7 is still unknown.
We also note that the quadratic forms associated to the remaining 56 pairs (cf. Tables 2, 3, 4, 6)
have the signature (3, 2) and the arithmeticity of Γ(f, g) associated to the 11 pairs of Table 2 follows
from Venkataramana [16]; and the arithmeticity of Γ(f, g) associated to the other 2 pairs of Table 3
follows from Singh [11]. Therefore, before this article arithmeticity of only 13 hypergeometric groups
Γ(f, g) ⊂ O(3, 2) was known and the arithmeticity or thinness of the remaining 43 hypergeometric
groups Γ(f, g) ⊂ O(3, 2) was unknown.
In this article, we show the arithmeticity of more than half of the remaining 43 hypergeometric
groups Γ(f, g) ⊂ O(3, 2). In fact, we obtain the following theorem:
Theorem 2. The hypergeometric groups associated to the 23 pairs of polynomials in Table 4 are
arithmetic.
Therefore, 23 of the remaining 43 pairs correspond to arithmetic hypergeometric groups; and
the arithmeticity or thinness of the hypergeometric groups associated to the remaining 20 pairs (cf.
Table 6) is still unknown.
Acknowledgements
We thank Max Planck Institute for Mathematics for the postdoctoral fellowships, and for providing us very pleasant hospitality. We also thank the Mathematisches Forschungsinstitut Oberwolfach

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JITENDRA BAJPAI AND SANDIP SINGH

where we met T.N. Venkataramana, and thank him for many discussions on the subject of the paper.
We also thank Wadim Zudilin for discussions at MPI.
2. Tables
In this section, we list all possible (up to scalar shifts) pairs f, g of degree five polynomials, which
are products of cyclotomic polynomials, and for which, the pair (f, g) form a primitive pair [1],
(0)
= −1, so that the associated monodromy group Γ(f, g) preserves a quadratic form Q, and
and fg(0)
Γ(f, g) ⊂ OQ (Z) as a Zariski dense subgroup (except for the 4 cases of Table 5). Note that once
we know the parameters α, β, the polynomials f, g can be determined by using the Equation (1.2).
Table 1. List of the 7 monodromy groups, for which, the associated quadratic form
Q has signature (4, 1), and thinness follows from Fuchs, Meiri, and Sarnak [5].
No.

α

1

0, 0, 0, 13 , 23

3

0, 0, 0, 14 , 34

5

0, 31 , 23 , 16 , 65

7

1
3
7
9
0, 10
, 10
, 10
, 10

β
1 1 3 5 7
2, 8, 8, 8, 8



1 1
3
7
9
2 , 10 , 10 , 10 , 10







1 1 2 3 4
2, 5, 5, 5, 5



1 1 3 5 7
2, 8, 8, 8, 8





No.

α

2

0, 0, 0, 31 , 32

4

0, 0, 0, 61 , 65

6

1
3
7
9
0, 10
, 10
, 10
, 10

β
1 1
5
7 11
2 , 12 , 12 , 12 , 12






1 1
3
7
9
2 , 10 , 10 , 10 , 10



1 1 2 1 3
2, 3, 3, 4, 4






Table 2. List of the 11 monodromy groups, for which, the associated quadratic
form Q has signature (3, 2), and arithmeticity follows from Venkataramana [16].
No.

α

8

0, 0, 0, 13 , 3

10

0, 31 , 13 , 23 , 32

12

0, 31 , 13 , 23 , 32

14

0, 31 , 23 , 14 , 43

16

3
7
9
1
, 10
, 10
, 10
0, 10

18

1
5
7 11
0, 12
, 12
, 12
, 12

β
1 1
3
7
9
2 , 10 , 10 , 10 , 10


2

1 1 2 3 4
2, 5, 5, 5, 5



1 1
3
7
9
2 , 10 , 10 , 10 , 10






1 1 2 3 4
2, 5, 5, 5, 5




1 1 1 3 3
2, 4, 4, 4, 4




1 1
3
7
9
2 , 10 , 10 , 10 , 10






No.

α

9

0, 13 , 31 , 23 , 3

11

0, 13 , 31 , 23 , 23

13

0, 13 , 31 , 23 , 23

15

0, 15 , 52 , 35 , 45

17

5
7 11
1
, 12
, 12
, 12
0, 12

β
1 1 1 3 3
2, 4, 4, 4, 4


2

1 1 1 5 5
2, 6, 6, 6, 6





1 1
5
7 11
2 , 12 , 12 , 12 , 12






1 1 1 3 3
2, 4, 4, 4, 4



1 1 1 3 3
2, 4, 4, 4, 4






ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

5

Table 3. List of the 2 monodromy groups, for which, the associated quadratic form
Q has signature (3, 2), and arithmeticity follows from Singh [11].
No.

α

19

No.

β

0, 0, 0, 0, 0



1 1 1 3
2, 4, 4, 4, 4


3

20

α

β

0, 0, 0, 0, 0

1 1 1 5 5
2, 6, 6, 6, 6





Table 4. List of the 23 monodromy groups, for which, the associated quadratic
form Q has signature (3, 2), and arithmeticity is shown in Section 3 of this paper.
No.

α

No.

β

2

21

0, 0, 0, 13 , 3

23

0, 0, 0, 13 , 23

25

0, 0, 0, 16 , 56

27

0, 31 , 13 , 23 , 32

29

0, 31 , 23 , 14 , 43

31

0, 31 , 23 , 16 , 65

33

0, 51 , 25 , 35 , 54

35

0, 61 , 16 , 56 , 65

37

0, 61 , 16 , 56 , 65

39

0, 81 , 38 , 58 , 87

41

1
5
7 11
0, 12
, 12
, 12
, 12

43

1
5
7 11
0, 12
, 12
, 12
, 12


3

1 1 1 3
2, 4, 4, 4, 4
1 1 1 5 5
2, 6, 6, 6, 6



1 1 1 3 3
2, 4, 4, 4, 4



1 1 3 5 7
2, 8, 8, 8, 8



0, 0, 0, 31 , 3



24

0, 0, 0, 41 , 43

26

0, 13 , 31 , 23 , 23



28

0, 13 , 32 , 14 , 34

30

0, 13 , 32 , 16 , 56

32

0, 13 , 32 , 16 , 56



34

0, 15 , 52 , 35 , 45



36

0, 16 , 61 , 56 , 56



38

0, 18 , 83 , 58 , 78



40

0, 18 , 83 , 58 , 78



42

1
5
7 11
0, 12
, 12
, 12
, 12



1 1
5
7 11
2 , 12 , 12 , 12 , 12



1 1 1 2 2
2, 3, 3, 3, 3



1 1 1 3 3
2, 4, 4, 4, 4

1 1 3 5 7
2, 8, 8, 8, 8



1 1 1 2 2
2, 3, 3, 3, 3



1 1 2 1 3
2, 3, 3, 4, 4




1 1 2 1 3
2, 3, 3, 4, 4
1 1 3 5 7
2, 8, 8, 8, 8

β

2

22




α







1 1 3 1 5
2, 4, 4, 6, 6
1 1 1 5 5
2, 6, 6, 6, 6





1 1 3 1 5
2, 4, 4, 6, 6



1 1 1 3 3
2, 4, 4, 4, 4







1 1 3 5 7
2, 8, 8, 8, 8



1 1
5
7 11
2 , 12 , 12 , 12 , 12



1 1 2 1 3
2, 3, 3, 4, 4



1 1 1 3 3
2, 4, 4, 4, 4



1 1 2 1 3
2, 3, 3, 4, 4



1 1 1 2 2
2, 3, 3, 3, 3



1 1 2 3 4
2, 5, 5, 5, 5












Table 5. List of the 4 monodromy groups, for which, the associated quadratic form
Q is positive definite (since the roots of the corresponding polynomials interlace on
the unit circle [1, Corollary 4.7]).
No.

α

44

0, 13 , 32 , 41 , 34

46

0, 15 , 52 , 53 , 45

β



1 1
3
7
9
2 , 10 , 10 , 10 , 10
1 1
3
7
9
2 , 10 , 10 , 10 , 10




No.

α

45

0, 13 , 32 , 16 , 56

47

0, 18 , 83 , 58 , 78

β



1 1
3
7
9
2 , 10 , 10 , 10 , 10
1 1
3
7
9
2 , 10 , 10 , 10 , 10




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JITENDRA BAJPAI AND SANDIP SINGH

Table 6. List of the 20 monodromy groups, for which, the associated quadratic
form Q has signature (3, 2), and arithmeticity or thinness is unknown.
No.

α

No.

β


1 1 1 1
2, 2, 2, 2, 2



1 1 1 2 2
2, 3, 3, 3, 3



1 1 2 3 4
2, 5, 5, 5, 5


1

48

0, 0, 0, 0, 0

50

0, 0, 0, 0, 0

52

0, 0, 0, 0, 0

54

0, 0, 0, 0, 0

56

0, 0, 0, 0, 0

58

0, 0, 0, 0, 0

60

0, 0, 0, 14 , 34

62

0, 0, 0, 16 , 56

64

0, 0, 0, 16 , 56

66

1
3
7
9
0, 10
, 10
, 10
, 10

1 1 3 5 7
2, 8, 8, 8, 8



53

0, 0, 0, 0, 0



55

0, 0, 0, 0, 0



57

0, 0, 0, 0, 0

59

0, 0, 0, 0, 0

61

0, 0, 0, 41 , 43



63

0, 0, 0, 61 , 65

65

0, 0, 0, 61 , 65



67

1
5
7 11
0, 12
, 12
, 12
, 12

3
7
9
1 1
2 , 10 , 10 , 10 , 10



1 1 1 1 2
2, 2, 2, 3, 3

1 1 1 2 2
2, 3, 3, 3, 3



0, 0, 0, 0, 0





1 1 2 1 3
2, 3, 3, 4, 4

51

1 1 2 1 5
2, 3, 3, 6, 6

1 1 2 1 3
2, 3, 3, 4, 4









1 1 1 1 2
2, 2, 2, 3, 3

0, 0, 0, 0, 0

1 1 1 1 3
2, 2, 2, 4, 4

1 1 1 1 2
2, 2, 2, 3, 3

β


49





α











1 1 1 1 5
2, 2, 2, 6, 6



1 1 3 1 5
2, 4, 4, 6, 6



5
7 11
1 1
2 , 12 , 12 , 12 , 12



1 1 1 2 2
2, 3, 3, 3, 3



1 1 1 2 2
2, 3, 3, 3, 3



1 1 2 3 4
2, 5, 5, 5, 5



1 1 1 2 2
2, 3, 3, 3, 3













Table 7. List of the 10 monodromy groups, for which, the associated quadratic
form Q has signature (4, 1), and arithmeticity or thinness is unknown.
No.

α

68

0, 0, 0, 31 , 3

70

0, 0, 0, 31 , 32

72

0, 0, 0, 41 , 43

74

0, 0, 0, 41 , 43

76

0, 0, 0, 61 , 65

β

2

1 1 1 1
2, 2, 2, 4, 4


3



1 1 2 3 4
2, 5, 5, 5, 5



5
7 11
1 1
2 , 12 , 12 , 12 , 12




1 1 2 3 4
2, 5, 5, 5, 5




1 1
5
7 11
2 , 12 , 12 , 12 , 12




No.

α

69

0, 0, 0, 13 , 3

71

0, 0, 0, 14 , 34

73

0, 0, 0, 14 , 34

75

0, 0, 0, 16 , 56

77

1
3
7
9
0, 10
, 10
, 10
, 10

β
1 1 1 1 5
2, 2, 2, 6, 6


2


1 1 2 1 5
2, 3, 3, 6, 6



1 1 3 5 7
2, 8, 8, 8, 8

1 1 3 5 7
2, 8, 8, 8, 8




1 1 2 3 4
2, 5, 5, 5, 5







ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

7

3. Proof of Theorem 2
In this section, we show the arithmeticity of all the hypergeometric groups associated to the
pairs of polynomials listed in Table 4, and it proves Theorem 2.
Strategy. We first compute the quadratic forms Q (up to scalar multiplications) preserved by the
hypergeometric groups of Theorem 2, and then show that the real rank of the orthogonal group
SOQ is two, and the Q - rank is either one or two. We then form a basis {ǫ1 , ǫ2 , ǫ3 , ǫ∗2 , ǫ∗1 } of Q5 ,
which satisfy the following condition: in Q - rank two cases, the matrix form of the quadratic form
Q, with respect to the basis {ǫ1 , ǫ2 , ǫ3 , ǫ∗2 , ǫ∗1 }, is anti-diagonal; and in Q - rank one cases, the vectors
ǫ1 , ǫ∗1 are Q - isotropic non-orthogonal vectors (that is, Q(ǫ1 , ǫ1 ) = Q(ǫ∗1 , ǫ∗1 ) = 0 and Q(ǫ1 , ǫ∗1 ) 6= 0),
and the vectors ǫ2 , ǫ3 , ǫ∗2 are Q - orthogonal to the vectors ǫ1 , ǫ∗1 .
Let P be the parabolic Q - subgroup of SOQ , which preserves the following flag:

{0} ⊂ Qǫ1 ⊂ Qǫ1 ⊕ Qǫ2 ⊕ Qǫ3 ⊕ Qǫ∗2 ⊂ Qǫ1 ⊕ Qǫ2 ⊕ Qǫ3 ⊕ Qǫ∗2 ⊕ Qǫ∗1
and U be the unipotent radical of P. It can be checked easily that the unipotent radical U is
isomorphic to Q3 (as a group), and in particular, U(Z) is a free abelian group isomorphic to Z3 .
We prove Theorem 2 (except for the cases 36, 39, and 42 of Table 4) by using the following
criterion of Raghunathan [8], and Venkataramana [15] (cf. [16, Theorem 6]): If Γ(f, g) ⊂ OQ (Z) is
Zariski dense and intersecting U(Z) in a finite index subgroup of U(Z), then Γ(f, g) has finite index
in OQ (Z). Note that, the criterion of Venkataramana [15] (for K - rank one) and Raghunathan
[8] (for K - rank two) are for all absolutely almost simple linear algebraic groups defined over a
number field K, and we have stated in a way to use it to prove our theorem.
To prove Theorem 2 for the cases 36, 39 and 42 of Table 4 (note that, for all these cases, the corresponding orthogonal groups have Q - rank two), we use the following criterion of Venkataramana
[14, Theorem 3.5]: If Γ(f, g) ⊂ OQ (Z) is a Zariski dense subgroup and intersecting the highest and
second highest root groups of SOQ non-trivially, then Γ(f, g) has finite index in OQ (Z).
For an easy reference to the root system, and the structures of the corresponding unipotent
subgroups of SOQ (of Q - rank two), we refer the reader to Remark 3.2 (cf. [11, Section 2.2]).
Notation. Let f, g be a pair of integer coefficient monic polynomials of degree 5, which have
roots of unity as their roots, do not have any common root, form a primitive pair, and satisfy
the condition: f (0) = −1 and g(0) = 1. Let A, B be the companion matrices of f, g respectively,
and Γ(f, g) be the group generated by A and B. Let C = A−1 B, I be the 5 × 5 identity matrix;
e1 , e2 , . . . , e5 be the standard basis vectors of Q5 over Q; and v be the last column vector of C − I,
that is, v = (C − I)e5 . Let Q be the non-degenerate quadratic form on Q5 , preserved by Γ(f, g).
Note that, Q is unique only up to scalars, and its existence follows from Beukers and Heckman [1].

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JITENDRA BAJPAI AND SANDIP SINGH

It is clear that C(v) = −v (since C 2 = I), and hence v is Q - orthogonal to the vectors e1 , e2 , e3 , e4
(since Q(ei , v) = Q(C(ei ), C(v)) = Q(ei , −v) = −Q(ei , v), for i = 1, 2, 3, 4); and Q(v, e5 ) 6= 0 (since
Q is non-degenerate). We may now assume that Q(v, e5 ) = 1.
It can be checked easily that the set {v, Av, A2 v, A3 v, A4 v} (similarly {v, Bv, B 2 v, B 3 v, B 4 v})
forms a basis of Q5 (cf. [11, Lemma 2.1]). Therefore, to determine the quadratic form Q on Q5 , it
is enough to compute Q(v, Aj v), for j = 0, 1, 2, 3, 4. Also, since v is Q - orthogonal to the vectors
e1 , e2 , e3 , e4 , and Q(v, e5 ) = 1 (say), we get

Q(v, Aj v) = coefficient of e5 in Aj v.

(3.1)

We compute the signature of the quadratic form Q by using [1, Theorem 4.5], which says the
following: By renumbering, we may assume that 0 ≤ α1 ≤ α2 ≤ . . . ≤ α5 < 1 and 0 ≤ β1 ≤ β2 ≤
. . . ≤ β5 < 1. Let mj = #{k : βk < αj } for j = 1, 2, 3, 4, 5. Then, the signature (p, q) of the
quadratic form Q is determined by the equation

(3.2)

|p − q| =

5
X

(−1)j+mj .

j=1

Now using the above equality with the non-degeneracy of the quadratic form Q (that is, p+q = 5),
one can check easily that all the quadratic forms Q associated to the pairs of polynomials in Tables
4 and 6 have signature (3, 2), that is, the associated orthogonal groups OQ have real rank two.
Therefore, the quadratic forms Q in the cases of Tables 4 and 6, have two linearly independent
isotropic vectors in Q5 (by the Hasse-Minkowski theorem), which are not Q - orthogonal, that is,
the associated orthogonal groups OQ have Q - rank ≥ 1.
Therefore, we are able to use the criterion of Raghunathan [8] and Venkataramana [15] (cf.
[16, Theorem 6]) to show the arithmeticity of the hypergeometric groups associated to the pairs
of polynomials of Table 4, and could not succeed to do the same for the hypergeometric groups
associated to the pairs of Table 6.
In the proof of arithmeticity of the hypergeometric groups associated to the pairs of polynomials
of Table 4, we denote by X the change of basis matrix (cf. second paragraph of Section 3), that is,
ǫ1 = X(e1 ),

ǫ2 = X(e2 ),

ǫ3 = X(e3 ),

ǫ∗2 = X(e4 ),

ǫ∗1 = X(e5 ).

We also denote by a, b the matrices A, B respectively, with respect to the new basis, that is,
a = X −1 AX,

b = X −1 BX.

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

9

We compute the quadratic forms Q on Q5 by using Equation (3.1), and denote by Q only, the
matrix forms of the quadratic forms Q, with respect to the standard basis {e1 , e2 , e3 , e4 , e5 } of Q5 ,
so that the conditions (note that At , B t denote the transpose of the respective matrices)

At QA = Q,

B t QB = Q

are satisfied.
Let P be the parabolic Q - subgroup of SOQ (defined in the third paragraph of Section 3),
and let U be the unipotent radical of P. Since U(Z) is a free abelian group isomorphic to Z3 ,
to show that Γ(f, g) ∩ U(Z) has finite index in U(Z), it is enough to show that it contains three
linearly independent vectors in Z3 . In the proof below, we denote the three corresponding unipotent
elements by q1 , q2 , and q3 (note that these are the words in a and b which are respectively some
conjugates of A and B (see the last paragraph), and hence (with respect to some basis of Q5 )
q1 , q2 , q3 ∈ Γ(f, g)); and in some of the Q - rank two cases, we are able to show only two unipotent
elements q1 and q2 inside Γ(f, g), which correspond, respectively, to the highest and second highest
roots (Remarks 3.2, 3.3, 3.4), and the arithmeticity of Γ(f, g) follows; thanks to the criterion [14,
Theorem 3.5] of Venkataramana.
We now note the following remarks:
Remark 3.1. As explained above, to show the arithmeticity of the hypergeometric groups we need
to produce enough unipotent elements in Γ(f, g). Note that in the case of symplectic hypergeometric
groups (when n is even and f (0) = g(0) = 1) the matrix C = A−1 B itself is a non-trivial unipotent
element, and to produce more unipotent elements in Γ(f, g), one can take the conjugates of C
and then the commutators of those conjugates, by keeping in mind the structure of the required
unipotent elements. But in case of the orthogonal hypergeometric groups, the element C = A−1 B
is not unipotent (since one of the eigen value of C is −1 and all other eigen values are 1) and
if we take C 2 (to get rid of the eigen value −1), it becomes the identity element. Therefore, to
find a non-trivial unipotent element in Γ(f, g), we take conjugates of C by the elements of Γ(f, g),
and then try to see the commutators of those conjugates. The idea of finding the first non-trivial
unipotent element is completely experiment based and we use Maple extensively to find it in some
of the cases. Once we find a non-trivial unipotent element, we take the conjugates of it and then
compute the commutators of such conjugates.
Remark 3.2. For the Q - rank two cases, the explicit structures of the unipotent subgroups
corresponding to the roots of the orthogonal group have been given in [11, Seciton 2.2]. For
pedagogical reasons, we summarize it here.
Note that if we consider the maximal torus
t1 0 0 0
 0 t2 0 0
0
T =  0 0 1 −1
 0 0 0 t2
0

0

0

0

0
0
0
0



 : ti ∈ Q ∗ , ∀ 1 ≤ i ≤ 2

−1

t1



10

JITENDRA BAJPAI AND SANDIP SINGH

inside the orthogonal group, and if ti is the character of T defined by
t
1
0
0

0
0

0
t2
0
0
0

0
0
1
0
0

0
0
0
t−1
2

0
0
0
0

0

t−1
1





 7−→ ti ,


for i = 1, 2,

−1 −1 −1 −1
−1
then the roots of the orthogonal group are given by Φ = {t1 , t2 , t1 t2 , t1 t−1
2 , t1 , t2 , t1 t2 , t2 t1 }.
−1
+
If we fix a set of simple roots ∆ = {t2 , t1 t−1
2 }, then the set of positive roots Φ = {t2 , t1 t2 , t1 , t1 t2 },
−1
−1 −1 −1 −1
−
the set of negative roots Φ = {t2 , t2 t1 , t1 , t1 t2 }, and t1 t2 , t1 are respectively the highest
and second highest roots in Φ+ .

The unipotent subgroups corresponding to the highest and the second highest roots are respectively given by



 1 0 0 x

λ
0 
1 0 x 0 − 2λ1 x2


3
 0 1 0 0 − λ1 x

 0 1 0 0

0
λ2


λ
Ut1 t2 = 0 0 1 0
and Ut1 = 0 0 1 0 − λ13 x  : x ∈ Q
0  : x ∈ Q
 0 0 0 1



0
0

 0 0 0 1
0

0

0

0

1

0

0

0

0

1

where λ1 =
λ2 =
and λ3 = Q(ǫ3 , ǫ3 ), and
is a basis of Q5 , with
respect to which, the matrix associated to the quadratic form Q has the anti-diagonal form.
Q(ǫ1 , ǫ∗1 ),

Q(ǫ2 , ǫ∗2 )

{ǫ1 , ǫ2 , ǫ3 , ǫ∗2 , ǫ∗1 }

Remark 3.3. Since Γ(f, g) is Zariski dense in the orthogonal group OQ (by [1, Theorem 6.5]),
to apply the criterion [14, Theorem 3.5] in the cases of hypergeometric groups of degree five, for
which, the corresponding orthogonal groups OQ have Q - rank two, it is enough to show that
[Ut1 t2 (Z) : Γ(f, g) ∩ Ut1 t2 (Z)]

and

[Ut1 (Z) : Γ(f, g) ∩ Ut1 (Z)]

are finite. Since
[U(Z) : Γ(f, g) ∩ U(Z)] < ∞ ⇒ [Ut1 t2 (Z) : Γ(f, g) ∩ Ut1 t2 (Z)] < ∞
and
[Ut1 (Z) : Γ(f, g) ∩ Ut1 (Z)] < ∞,
the arithmeticity of the hypergeometric groups in Table 4, for which, the associated orthogonal
groups have Q - rank two, also follows from [14, Theorem 3.5].
Remark 3.4. In the three cases 36, 39 and 42, we find the unipotent elements (in Γ(f, g)) corresponding to the highest and the second highest roots, and finding the third unipotent element in
these cases does not seem to be direct compare to the other 20 cases of Table 4, so we do not put
much effort in computing the third unipotent element (to show that [U(Z) : Γ(f, g) ∩ U(Z)] < ∞).
Also, once we find that the group Γ(f, g) has finite index in the integral orthogonal group (using
[14, Theorem 3.5]) it follows trivially that Γ(f, g) contains the unipotent elements corresponding
to each roots, and hence [U(Z) : Γ(f, g) ∩ U(Z)] is also finite.
Remark 3.5. In some of the remaining 20 cases (cf. Table 6), for which, the associated quadratic
forms Q have signature (3, 2), we find some non-trivial unipotent elements (but not enough to show
the arithmeticity), and for the other cases, it is even not possible (for us) to find a non-trivial
unipotent element to start with. Thus, we are not able to show the arithmeticity of these groups,
and believe that one may apply the ping-pong argument (similar to Brav and Thomas [2]) to show
the thinness of these groups.

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

11

We now give a detailed explanation of the computation for finding out the unipotent elements
inside the hypergeometric group Γ(f, g) in the first case (Case 21) of Table 4. Following the similar
computation, we find out the unipotent elements for the other cases of Table 4.
3.1. Arithmeticity of Case 21. In this case


1 2
;
α = 0, 0, 0, ,
3 3

β=

f (x) = x5 − 2x4 + x3 − x2 + 2x − 1;



1 1 1 3 3
, , , ,
2 4 4 4 4



g(x) = x5 + x4 + 2x3 + 2x2 + x + 1.

Let A and B be the companion matrices of f (X) and g(X) respectively, and let C = A−1 B.
Then








A=


0

0

0

0

1

0

0

0

0

−1

1

0

0

0

−3

1

0

0

0

−2

1

0

0

0

−1

0

1

0

0

−1

0

1

0

0

1

0

1

0

0

−2

0

0

1

0

−3

0

0

1

0

−1

0

0

0

1

1

0

0

0

1

2

0

0

0

0

−1



,




B=


0

0

1

0

−2

0

0

0

1

−1



,




C=




.


Let {e1 , e2 , e3 , e4 , e5 } be the standard basis of Q5 over Q, and let v = (C − I)(e5 ) = −3e1 −
e2 − 3e3 + e4 − 2e5 . Then Bv = 2e1 − e2 + 3e3 + e4 + 3e5 , B 2 v = −3e1 − e2 − 7e3 − 3e4 − 2e5 ,
B 3 v = 2e1 − e2 + 3e3 − 3e4 − e5 , B 4 v = e1 + 3e2 + e3 + 5e4 − 2e5 ; and hence by the Equation (3.1)
it follows that, with respect to the basis {v, Bv, B 2 v, B 3 v, B 4 v}, the matrix of the quadratic form
(up to scalar multiplication) preserved by the hypergeometric group Γ(f, g) is given by




Q=


−2

3

−2

−1

−2

3

−2

3

−2

−1

−2

3

−2

3

−2

−1

−2

3

−2

3

−2

−1

−2

3

−2



.


We now let Z be the change of basis matrix from the basis {v, Bv, B 2 v, B 3 v, B 4 v} to the standard
basis {e1 , e2 , e3 , e4 , e5 }. Then, a computation shows that




Z=


−5/32

−11/32

3/32

5/32

1/2

−1/2

−1/4

1/4

0

7/16

−3/16

−5/16

1/16

−1/16

−5/32

1/4

−1/4

0

0

−1/4

11/32

−3/32

−5/32

5/32

−5/32






and, with respect to the standard basis {e1 , e2 , e3 , e4 , e5 }, the matrix (up to scalar multiplication)
of the quadratic form preserved by the hypergeometric group Γ(f, g) is given by




−32Z t QZ = 


5

−5

−3

11

5

−5

5

−5

−3

11

−3

−5

5

−5

−3

11

−3

−5

5

−5

5

11

−3

−5

5






12

JITENDRA BAJPAI AND SANDIP SINGH

(here we multiply by −32, just to get rid of the denominators) where Z t denotes the transpose of
the matrix Z. Thus, we find that
At (−32Z t QZ)A = (−32Z t QZ) = B t (−32Z t QZ)B.
After computing the isotropic vectors of the quadratic form Q, we change the standard basis
{e1 , e2 , e3 , e4 , e5 } to the basis {ǫ1 , ǫ2 , ǫ3 , ǫ∗2 , ǫ∗1 }, for which, the change of basis matrix is given by


4



X=


0

0

0

−1

4

0

−2

6

2

4

4

−2

2

−2

4

4

−6

10

0

0

0

2

−2

−3



.


With respect to the new basis, the matrices of Q, A and B are X t QX, a = X −1 AX and b =
X −1 BX; which are respectively










0

0

0

0

0
0

0

0

−128

0

64

0

0

−128

0

0

−128

0

0

0

 0

 2


 4


 2

−128 

0 


0 ,


0 

0

0

1/2

−1/2
−1

−1

3

0

−3

6

1

−2

3

−2

0

0

 0

 2


 4


 2

0 

2 


2 ,


1/2 

−1/2

0

3

0

−1/2

0

−1

−1

3

0

−3

6

1

−2

3

−2

2

−2

3/4 

2 


2 .


1/2 
0

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
w2 = a−1 b4 a,

w1 = ab−1 ab,
then



−1

−1

2

−3

−1

0

−1

2

−4

−1

0

0

1

−4

0



w1 = 





w3 = 


w3 = w12 w2−1 ,

0

0

0

−1

1

0

0

0

0

−1

1

0

−2

2

4

0

1

0

0

−2

0

0

1

0

−4

0

0

0

1

0

0

0

0

0

1

and the final three unipotent elements are







w2 = 




,








,




w4 = 


q2 = w32 q1 ,

q1 = w4 w2 ,

w4 = (w3 w1−1 )2 w3 ,
1

2

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

−2

0

0

0

0

1

1

−2

0

−4

−8

0

1

0

0

4

0

0

1

0

0

0

0

0

1

2

0

0

0

0

1





,





,


q3 = w2 ,

which are respectively
 1

 0


 0


 0
0

0

0

−4

1

0

0

0

1

0

0

0

1

0

0

0

0 

4 


0 ,


0 
1

 1

 0


 0


 0
0

0

−4

0

1

0

0

0

1

0

0

0

1

0

0

0

16 

0 


−8  ,


0 
1

 1

 0


 0


 0

0

2

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0 

0 


0 .


−2 
1

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

13

It is clear from the above computation that the group generated by q1 , q2 , q3 is a subgroup of
Γ(f, g) ∩ U(Z), and has finite index in U(Z), the integer points of the unipotent radical U of the
parabolic subgroup P, which preserves the flag
{0} ⊂ Qǫ1 ⊂ Qǫ1 ⊕ Qǫ2 ⊕ Qǫ3 ⊕ Qǫ∗2 ⊂ Qǫ1 ⊕ Qǫ2 ⊕ Qǫ3 ⊕ Qǫ∗2 ⊕ Qǫ∗1 .
Therefore Γ(f, g)∩U(Z) is of finite index in U(Z). Since Γ(f, g) is also Zariski dense in the orthogonal
group OQ (by Beukers and Heckman [1]), the arithmeticity of Γ(f, g) follows from Venkataramana
[15] (cf. [8], [16, Theorem 6]).
Remark 3.6. We follow the same notation as that of Section 3.1 for the remaining cases, and the
arithmeticity of Γ(f, g) follows by the similar arguments used in Section 3.1. Also, in the proof of
the arithmeticity of the remaining cases, we denote the commutator of two elements s, t (say) in
Γ(f, g) by [s, t] which is sts−1 t−1 .
3.2. Arithmeticity of Case 22. In this case


1 2
;
α = 0, 0, 0, ,
3 3

β=

f (x) = x5 − 2x4 + x3 − x2 + 2x − 1;



1 1 3 1 5
, , , ,
2 4 4 6 6



g(x) = x5 + x3 + x2 + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
1

3



1
Q=


−5



−7

3

1

−5

1

3

1

3

1

3

1

3

1

−5

1

3

−7 

−5 


1 ;


3 
1

0

1



0
X =


 −1


0

2

−2

−1/2

0

0

0

4

−2

0

0

−2

−3

2

−2

−1/2

1 

−1/2 


2 ,


−1/2 
0

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively
 0

 0


 0


 0

0

0

0

0

0

−16

0

−16

0

−16

0

0

0

0

0

8

8 

0 


0 ,


0 
0

 1/2

 1/4


1



 −1
1

−3

4

9/4

1/2

−1

−9/8

−2

3

3/2

2

−4

−5/2

−2

4

7/2

 1/2

 1/4


1



 −1

5/4 

−1/8 


1/2  ,


−3/2 

1

1/2

1

0

5/4

3/2

−2

−11/8

2

−1

1/2

−2

0

−3/2

−2

4

7/2

5/4 

−1/8 


1/2  .


−3/2 
1/2

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
w1 = [a−1 , b], w2 = (aba)2 (bab)−1 , w3 = [w22 , w1 ],
then








w1 = 


1

4

−4

−1

2

−1

0

0

1

0

1

0

−8

−16

16

0

1

0

0

−1/2

0

−1

−1

1/2

1/2

0

1

0

0

−8

0

0

1

0

−2

−2

0

1

1

0

0

0

1

0

−4

0

0

0

1

2

0

0

0

1

0

0

0

0

0

1

0

0

0

0

1



,




w2 = 


0

0

0

−1

0

−4

0

4

0

−1



,




w3 = 




,


14

JITENDRA BAJPAI AND SANDIP SINGH

and the final three unipotent elements are
q1 = [w22 , w3 ],

q3 = (w1−4 q22 )4 q1 ,

q2 = w3 q1−1 ,

which are respectively
 1

 0


 0


 0
0

0

0

−16

1

0

0

0

1

0

0

0

1

0

0

0

 1

 0


 0


 0

0 

−8 


0 ,


0 

0

1

0

−8

0

1

0

0

0

1

0

0

0

1

0

0

0

3.3. Arithmeticity of Case 23. In this case


1 2
α = 0, 0, 0, ,
;
3 3



−64

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

1

β=

f (x) = x5 − 2x4 + x3 − x2 + 2x − 1;

 1

 0


 0


 0

16 

0 


−4  ,


0 

1 1 1 5 5
, , , ,
2 6 6 6 6

0 

0 


0 .


−32 
1



g(x) = x5 − x4 + x3 + x2 − x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
3

3



1
Q=


 −3

3

1

−3

3

3

1

3

3

3

1

3

3

−3

1

3



−7

0

 −2

0



0
X =


 −2

−7 

−3 


1 ;


3 

0

3

0

0

−2

8

−1

0

−8

2

0

8

−4

−2

0

1

1 

−3 


2 ,


−1 
1

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively
0

0

0

0

16

0

−64

0

0

16

0

0

16

0

0

0











0

0

0
0

 0

 1


 0


 0

16 

0 


0 ,


0 
0

0

0

0

−1/2

1

−4

1

0

−1

0

0

0

0

−2

0

0

1/2 

1 


1 ,


1 
2

 0

 1


 0


 0
0

−1

0

0

1

−4

1

0

−1

0

0

0

0

0

0

−1

1 

1 


1 .


1 
1

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
w1 = a−1 b6 a,

w2 = ab−6 a−1 ,

w5 = [a−1 , b]a−2 ,

w6 = dw1 d,

d = ba−1 ,

w3 = (ab−4 )2 ,

w4 = (a−1 b4 )2 ,

w7 = w6−4 w5 w1 w5−1 w2 ,

then




d=


0

0

0

0

1/2

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

2

0

0

0

0





,






w1 = 


1

4

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

−4

0

0

0

0

1





,






w2 = 


1

0

0

0

0

4

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

−4

1





,


ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE





w3 = 


1

0

0

−1

0

2

1

0

−2

1

0

0

1

0

0

0

0

0

1

0

0

0

0

−2





1





w4 = 




,




w6 = 


1

2

0

1

−2

0

1

0

0

−1

0

0

1

0

0

0

0

0

1

−2

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

−8

0

0

1

0

0

8

0

0

1

and the final three unipotent elements are



q1 = w42 w1−1 ,



,


1





,








w7 = 




w5 = 


0

1

0

0

0

−1

1

0

0

0

−2

0

1

0

0

−8

−4

8

0

−1

4

−8

1

1

4

1

0

0

0

0

0

1

0

0

0

−8

0

1

0

0

0

0

0

1

0

128

0

−32

0

1

4

0

0

1

0

0

0

1

0

0

0

1

0

0

0

q2 = dw7 d,

15







,




,


q3 = w1 ,

which are respectively
 1

 0


 0


 0
0

0

0

2

1

0

0

0

1

0

0

0

1

0

0

0

0 

−2 


0 ,


0 
1

 1

 0


 0


 0
0

0

−16

0

1

0

0

0

1

0

0

0

1

0

0

0

3.4. Arithmeticity of Case 24. In this case


1 3
;
α = 0, 0, 0, ,
4 4

β=

f (x) = x5 − 3x4 + 4x3 − 4x2 + 3x − 1;

1

 1

 0


 0


 0



1 1 1 5 5
, , , ,
2 6 6 6 6

32 

0 


−4  ,


0 

0

0 

0 


0 .


−4 
1



g(x) = x5 − x4 + x3 + x2 − x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
7







Q=




9

7

−7

9

7

9

7

7

9

7

9

−7

7

9

7

−25

−7

7

9

−25 

−7 


7 ;


9 
7

0

 −1


0
X =


1



0

2

0

1

−2

0

−1

4

4

3

−6

−4

−2

2

0

1

−2 

3 


−6  ,


4 

−1

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0

0

0

−32

0

0

−64

0

0

−32

0

0

16

0

0

0

16 

0 


0 ,


0 
0

1

0

−4

1


 1/2


0



1


0

−2

0



1

0

−1

0

−2

−4

1

−2

−4

0

2 

1 


1 ,


1 

2

1

−4

−4

−1


 1/2


0



1


−1

−2

−1/2



1

0

−1

0

−4

−4

0

−2

−4

0

4 

3/2 


1 .


2 
2

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
c = a−1 b,

d = [c, a],

e = [a−1 , b2 ],

s = (cac)4 ,

w1 = e−1 de−1 d−1 ,

w2 = d[e, c]d−1 ,

16

JITENDRA BAJPAI AND SANDIP SINGH

w3 = w1−2 d−1 w2 sw2−1 d,

w4 = w3−1 sd−4 ,

w7 = d−82 (w4 dw4−1 )2 ,

w5 = [w4 , d],

w6 = a4 w4 dw4−1 d7 a−4

w8 = w62 (w4 dw4−1 d7 )−10 ,

then the final three unipotent elements are
q3 = (w5−2 w7 )288 q1 ,

q2 = w7−240 q3 ,

q1 = [w6 , w8 ],
which are respectively
 1

 0


 0


 0
0

0

0

−23040

1

0

0

0

1

0

0

0

1

0

0

0

0 

−11520 


0 ,


0 
1

 1

 0


 0


 0
0

0

23040

0

1

0

0

0

1

0

0

0

1

0

0

0

3.5. Arithmeticity of Case 25. In this case


1 5
α = 0, 0, 0, ,
;
6 6

66355200 

0 


5760  ,


0 
1

β=

f (x) = x5 − 4x4 + 7x3 − 7x2 + 4x − 1;



 1

 0


 0


 0

46080

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

1 1 1 3 3
, , , ,
2 4 4 4 4

0 

0 


0 .


23040 
1



g(x) = x5 + x4 + 2x3 + 2x2 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
37

11



Q =  −35


 −37

11

−35

−37

37

11

−35

11

37

11

−35

11

37

−37

−35

11



37

37 

−37 


−35  ;


11 
37

 4

 4


X = 4


 4
0

0

−24

−10

2

−24

−9

2

−48

−19

2

−24

−7

2

−24

−11

11 

6 


18  ,


8 
9

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

192

0

2304

0

0

192

0

0

−384

0

0

0

−384 

0 


0 ,


0 
0

 0

 2


 0


 0
0

−6

66

121/4

−10

108

105/2

−2

23

11

0

0

0

−2

24

12

−47/2 

−39 


−8  ,


−1 
−9

 0

 2


 0


 0
0

−1/2

0

0

0

−12

−5/2

0

−1

0

0

0

0

0

0

1

5/4 

6 


1 .


−1 
0

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
w1 = b4 , w2 = a−1 b4 a, w3 = w2 w1−12 , w4 = [a, b4 ],
w5 = (ab)−1 b−2 (ab)−1 b2 ,

w6 = w1−1 w2 w3−1 w4−1 w5−1 ,

then the final three unipotent elements are
q1 = w1 ,

q2 = w5−1 w4−1 w2 w6−1 w2−1 q1−277 ,

q3 = w2−32 q1384 q28 ,

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

17

which are respectively
 1

 0


 0


 0
0

0

0

1

1

0

0

0

1

0

0

0

1

0

0

0

 1

 0


 0


 0

0 

2 


0 ,


0 

0

1

0

96

0

1

0

0

0

1

0

0

0

1

0

0

0

 1

 0


 0


 0

768 

0 


16  ,


0 

β=

f (x) = x5 + x4 + x3 − x2 − x − 1;



0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

1

3.6. Arithmeticity of Case 26. In this case


1 1 2 2
α = 0, , , ,
;
3 3 3 3

64

1 1 3 1 5
, , , ,
2 4 4 6 6

0 

0 


0 .


128 
1



g(x) = x5 + x3 + x2 + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
1

0



Q =  −2


1


0



2

−2

1

1

0

−2

0

1

0

−2

0

1

1

−2

0

2 

1 


−2  ;


0 

 −2

0



X =  −4


 −2

1

0

0

4

0

2

0

1

0

4

0

4

0

0

2

−4

−1

−1 

1 


0 ,


1 
3

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively
 0

 0


 0


 0
8

0

0

0

0

0

−8

0

−16

0

−8

0

0

0

0

0

8 

0 


0 ,


0 
0

0

 −1


0



0


1



−2

−1

−1

2

0

2

−3

−1

0

0

0

4

−4

−1

0

1 

−1 


2 ,


1 
3

0

 −1


0



0



0

0

0

−1/2

−1

2

0

0

1

0

0

0

0

2

0

0

−1/2 

−1 


−1  .


1 
0

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
c = a−1 b, e = [a2 , b2 ], u = [a−1 , b2 ],

w = [a−1 , b4 ],

w2 = w1 (t2 a3 )−2 , w3 = w2 (w2−1 a−3 w2 )2 w1−1 ,
then the final three unipotent elements are
q1 = w4−4 q2 ,

t = w−1 ue−1 , w1 = (t2 a3 )−2 t4 (a3 (b4 a−1 )3 )2 ,

w4 = w3−2 w−4 w3 ,

q2 = w6 w4−32 w3−81 ,

w5 = cw1 c,

q 3 = w3 ,

which are respectively
 1

 0


 0


 0
0

0

0

16

1

0

0

0

1

0

0

0

1

0

0

0

0 

16 


0 ,


0 
1

 1

 0


 0


 0
0

0

−64

0

1

0

0

0

1

0

0

0

1

0

0

0

1024 

0 


−32  ,


0 
1

 1

 0


 0


 0
0

8

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0 

0 


0 .


8 
1

w6 = w5 w3 w5−1 ,

18

JITENDRA BAJPAI AND SANDIP SINGH

3.7. Arithmeticity of Case 27. In this case


1 1 2 2
;
α = 0, , , ,
3 3 3 3

β=

f (x) = x5 + x4 + x3 − x2 − x − 1;



1 1 3 5 7
, , , ,
2 8 8 8 8



g(x) = x5 + x4 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
7

 −5


Q =  −1


7



−5

−1

7

7

−5

−1

−5

7

−5

−1

−5

7

7

−1

−5

−9

 1

 2


X = 1


 0

−9 

7 


−1  ;


−5 
7

0

0

0

0

0

4

−1

1

4

0

2

4

0

1

0

1

1 

0 


0 ,


−1 
0

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively
0

0

0

0

−4

0

−16

0

0

−4

0

0

−4

0

0

0











0

0

0
0

 −1

 −1


1



1


−4 

0 


0 ,


0 
0

1

1

0

0

0

0

0

0

1

0

1

4

−1

0

0

1

0 

0 


0 ,


−1 
0

 −1

 −1


1



1

1

0

0

−1

0

0

0

0

1

0

1

4

−1

−1

0

0

0 

0 


0 .


−1 
0

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
d = (ab−4 )2 ,

w1 = (a2 b4 a−1 d−1 )−2 ,
w5 = w2 w42 ,

w2 = (a−1 ba−4 b5 )2 ,
w6 = w34 w5 w34 w5−1 ,

w3 = [a−1 , b3 ],

w4 = w1 w3−1 w1−1

w7 = [w34 , w5 ],

then the final three unipotent elements are
q1 = [w7 , w5−1 ],

q2 = w7−2 q1 ,

q3 = w6−2 q2 ,

which are respectively
 1

 0


 0


 0
0

0

0

−16

1

0

0

0

1

0

0

0

1

0

0

0

0 

16 


0 ,


0 
1

 1

 0


 0


 0
0

0

32

0

1

0

0

0

1

0

0

0

1

0

0

0

3.8. Arithmeticity of Case 28. In this case


1 2 1 3
;
α = 0, , , ,
3 3 4 4
f (x) = x5 + x3 − x2 − 1;

−128 

0 


−8  ,


0 
1

β=



 1

 0


 0


 0
0

16

0

0

1

0

0

0

1

0

0

0

1

0

0

0

1 1 3 5 7
, , , ,
2 8 8 8 8



g(x) = x5 + x4 + x + 1.

0 

0 


0 .


−16 
1

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

19

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
3

 −3


Q =  −1


5



−5

−3

−1

5

3

−3

−1

−3

3

−3

−1

−3

3

5

−1

−3

 0

 0


X = 2


 2

−5 

5 


−1  ;


−3 
3

0

0

4/3

1

6

4/3

0

6

0

−1

10

−8/9

−1/3

−6

0

1

−2 

−1 


0 ,


0 

−3

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

48

0

−64/9

0

0

48

0

0

16

0

0

0

16 

0 


0 ,


0 
0

 −1/2

1/6



−3/2



0

−1

−7

−1/9

7/12

4/3

1/27

−7/36

−12

2/3

7/4

−6

−4/3

−1

−8

−2/9

1/6

−11/4 

19/36 


−17/4  ,


5/3 
−1/2

 −1/2

1/6



−3/2



0

−1

−1

−1/9

−5/12

1/3

1/27

−1/36

−3

2/3

1/4

−6

−4/3

−1

−8

−2/9

1/6

1/4 

1/36 


1/4  .


5/3 

−1/2

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
c = a−1 b,

e = ab−1 ,

w1 = a2 b4 a−2 ,

w2 = w1−1 b4 w1 ,

w5 = [w2 , w42 ], w6 = [c, w4−1 ],
then the final three unipotent elements are
q1 = w5−9 q2 ,

w3 = a−1 b4 a,

w7 = w6−16 w53 ,

q2 = w824 q3 ,

w4 = w3 b6 w3 ,

w8 = ew6 e,

q3 = w7 w8−8 ,

which are respectively
 1

 0


 0


 0
0

0

0

144

1

0

0

0

1

0

0

0

1

0

0

0

0 

−48 


0 ,


0 
1

 1

 0


 0


 0
0

0

96

0

1

0

0

0

1

0

0

0

1

0

0

0

10368 

0 


216  ,


0 
1

 1

 0


 0


 0
0

−432

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0 

0 


0 .


144 
1

3.9. Arithmeticity of Case 29 (Q - rank one case). In this case




1 2 1 3
1 1 5 7 11
α = 0, , , ,
, , , ,
;
β=
3 3 4 4
2 12 12 12 12
f (x) = x5 + x3 − x2 − 1;

g(x) = x5 + x4 − x3 − x2 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
1

 −3


Q =  −1


3



−5

−3

−1

3

1

−3

−1

−3

1

−3

−1

−3

1

3

−1

−3

−5 

3 


−1  ;


−3 
1

 −1

0



X =  −1


0

0

0

0

0

−1

−2

1

−2

−1

−1

−4

1

−1

1

2

−1

−1 

−1 


1 ,


2 
−1

20

JITENDRA BAJPAI AND SANDIP SINGH

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0

−24

−12

−12

0

−12

−24

12

0

−12

12

−8

−12

0

0

0

−12 

0 


0 ,


0 
0

 −1/2

1/2



0



0


1/2

−3

−3/2

1

1/2

1/6

0

0

−1/3

1/2

3

0

0

2

−1/2

1/2

3/2 

−1/2 


1 ,


0 

−1/2

 −1/2

1/2



0



0


−2

1/2

1

1/2

1/6

0

0

−1/3

1/2

−1/2

3

0

0

3

3/2

−1/2

1/2 

−1/2 


1 .


0 

−3/2

A computation shows that the Q - rank of the orthogonal group SOQ is one. If we denote by
c = a−1 b,

d = aba,

e = [a−3 , b−1 ],

w2 = a3 e−1 a−3 dca−3 ,

w1 = cdca−3 ,

w4 = w3 w2−1 , w5 = w3−1 w43 ,
then the final three unipotent elements are
q1 = w18 q2 ,

w6 = [e, w12 ],

q2 = w14 w7 ,

w3 = w1−2 w23 ,

w7 = w53 w6 ,

q3 = w34 q1−1 ,

which are respectively
 1

 0


 0


 0
0

0

0

24

1

0

0

0

1

0

0

0

1

0

0

0

216 

−18 


18  ,


18 
1

 1

 0


 0


 0

0

0

24

0

1

0

0

0

1

0

0

0

1

0

0

0

3.10. Arithmeticity of Case 30. In this case


1 2 1 5
α = 0, , , ,
;
3 3 6 6
f (x) = x5 − x4 + x3 − x2 + x − 1;

24 

−10 


2 ,


18 
1

β=



 1

 0


 0


 0

24

0

0

1

0

0

0

1

0

0

0

0

1

0

0

0

1 1 1 3 3
, , , ,
2 4 4 4 4

24 

2 


−10  .


−18 
1



g(x) = x5 + x4 + 2x3 + 2x2 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
1

 −7


1
Q=


 17


1

−7

1

17

1

−7

1

−7

1

−7

1

−7

1

17

1

−7

1 

17 


1 ;


−7 
1

0

1



0
X =


 −1


0

1

0

−1

1

0

−1

2

0

−1

3

−24

4

1

0

−1

2 

1 


4 ,


0 
1

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

−24

0

576

0

0

−24

0

0

−24

0

0

0

−24 

0 


0 ,


0 
0

1

 −1


0



1



1

−4

24

−3

7

−48

7

1

−7

1

0

0

0

−3

24

−4

0 

1 


0 ,


0 

0

1

 −1


0



1



1

−2

24

−5

6

−48

8

1

−7

1

1

0

−1

−3

24

−4

2 

0 


0 .


1 

0

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

21

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
d = [a, b−1 ],

e = [b−1 , a2 ],

s = ba2 b−1 ,

r = aba−1 ,

2

w5 = d2 w12 d−1 ,

w2 = [e−1 , t−1 ] , w3 = d−12 w2 , w4 = sw2 s−1 ,
then the final three unipotent elements are
q1 = w3 ,

t = [s, d−1 ],

q2 = [w6 , w2 ]w3−1728 ,

w1 = b−1 a−1 b,

w6 = w4 w52 w4−1 w52 ,

q 3 = w2 ,

which are respectively
 1

 0


 0


 0
0

0

0

−12

1

0

0

0

1

0

0

0

1

0

0

0

 1

 0


 0


 0

0 

12 


0 ,


0 

0

1

0

−3456

0

1

0

0

0

1

0

0

0

1

0

0

0

3.11. Arithmeticity of Case 31. In this case


1 2 1 5
;
α = 0, , , ,
3 3 6 6

248832 

0 


−144  ,


0 
1

β=

f (x) = x5 − x4 + x3 − x2 + x − 1;



 1

 0


 0


 0

−12

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

1 1 3 5 7
, , , ,
2 8 8 8 8

0 

0 


0 .


12 
1



g(x) = x5 + x4 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
1

5



1
Q=


 −7

5

1

−7

1

5

1

5

1

5

1

5

1

−7

1

5



1

1 

−7 


1 ;


5 
1

0

 −1


0
X =


1



0

2

0

−1

2

0

−1

2

0

0

−8

−12

−1

2

0

−1

2 

1 


0 ,


−2 
3

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

−24

0

144

0

0

−24

0

0

12

0

0

0

12 

0 


0 ,


0 
0



−1

 −1/2


0



1

1

10

12

0

2

0

−1

2

5

1

−14

−24

−3

−8

−12

−1

6 

2 


0 ,


−3 

−2



−1

 −1/2


0



1

1

6

12

2

1

0

−1/2

2

5

1

−12

−24

−4

−8

−12

−1

0 

1/2 


0 .


0 
−2

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
2

d = [b−1 , a] ,

w1 = ab4 a−1 ,

w2 = [a2 , w1−1 ],

w3 = w2 dw2−1 d,

then the final three unipotent elements are
q1 = w42 q2−1 ,

q2 = w2−1 w4 w2 w4 ,

q3 = (q1 d6 )2 q1−1 ,

w4 = w3 d−2 ,

22

JITENDRA BAJPAI AND SANDIP SINGH

which are respectively
 1

 0


 0


 0
0

0

0

24

1

0

0

0

1

0

0

0

1

0

0

0

0 

12 


0 ,


0 
1

 1

 0


 0


 0
0

0

−48

0

1

0

0

0

1

0

0

0

1

0

0

0

−96 

0 


4 ,


0 
1

 1

 0


 0


 0

48

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

0 

0 


0 .


24 
1

3.12. Arithmeticity of Case 32. In this case




1 1 5 7 11
1 2 1 5
, , , ,
;
β=
α = 0, , , ,
3 3 6 6
2 12 12 12 12
f (x) = x5 − x4 + x3 − x2 + x − 1;

g(x) = x5 + x4 − x3 − x2 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
1

2



1
Q=


−1



1

2

1

−1

1

2

1

2

1

2

1

2

1

−1

1

2

1 

−1 


1 ;


2 
1

0

 −1


0
X =


1



0

1

0

−1/2

1

0

−1/2

1

0

1/2

−4

−3

0

1

0

−1/2

1 

1/2 


−1  ,


−3/2 
2

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively
 0

 0


 0


 0
3

0

0

0

0

0

−3

0

9

0

−3

0

0

0

0

0

3 

0 


0 ,


0 
0

 −1/2

 −1/2


0



1

1

3

3/2

−1/2

3/2

0

−3/4

1

2

1/2

−7

−6

−1/2

−4

−3

0

15/4 

9/4 


0 ,


−5/2 
−3/2

 −1/2

 −1/2


0



1

1

1

3/2

1/2

1/2

0

−1/4

1

2

1/2

−5

−6

−3/2

−4

−3

0

−1/4 

1/4 


0 .


3/2 

−3/2

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
d = [b−1 , a], e = ab6 a−1 , w1 = [a2 , e−1 ], w2 = w1 dw1−1 d,
then the final three unipotent elements are


2 −1 2
−1
q2 , q2 = w1−1 w2 d−2 w1 w2 d−2 ,
q1 = q2 dq2 w2

q3 = (q2−1 dq2 )−96 q1 ,

which are respectively
 1

 0


 0


 0
0

0

0

−96

1

0

0

0

1

0

0

0

1

0

0

0

0 

−96 


0 ,


0 
1

 1

 0


 0


 0
0

0

−48

0

1

0

0

0

1

0

0

0

1

0

0

0

−384 

0 


16  ,


0 
1

 1

 0


 0


 0
0

−192

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0 

0 


0 .


−192 
1

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

3.13. Arithmeticity of Case 33. In this case


1 2 3 4
α = 0, , , ,
;
5 5 5 5
f (x) = x5 − 1;

β=



1 1 1 2 2
, , , ,
2 3 3 3 3

23



g(x) = x5 + 3x4 + 5x3 + 5x2 + 3x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
29

 −25


11
Q=


11



−25

11

11

29

−25

11

−25

29

−25

11

−25

29

11

11

−25

−25

−25 

11 


11  ;


−25 
29

 −1

 −1


1
X =


1

0

−2

2

3

−1

2

4

0

0

0

−1

0

−1

−2

2

0

6 

4 


1 ,


2 
5

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0

0

0

−18

0

0

36

0

0

−18

0

0

18

0

0

0

18 

0 


0 ,


0 
0

0

 −3


0



0


0

0

3

−2

4

4

0

−1

1

0

0

1

−1

2

1



−1

2 

4 


0 ,


1 

2

0

 −3


0



0


12



−1

−12

3

−8

10

4

2

−3

1

4

−4

1

−3

4

1

−28 

19 


−5  .


−9 
7

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
w1 = [a−1 , b−6 ],

w2 = [a−1 , b3 ],

w3 = a−3 b−6 a3 ,

w4 = w22 w1−1 w3 w2−2 w1 w3−1 ,

w5 = w4−2 w1 w4−1 w2−2 ,
then the final three unipotent elements are
q2 = w53 w410 ,

q1 = w3 ,

q3 = w4 ,

which are respectively
 1

 0


 0


 0
0

0

0

6

1

0

0

0

1

0

0

0

1

0

0

0

0 

6 


0 ,


0 
1

 1

 0


 0


 0
0

0

24

0

1

0

0

0

1

0

0

0

1

0

0

0

−144 

0 


−12  ,


0 
1

 1

 0


 0


 0
0

−48

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0 

0 


0 .


−48 

3.14. Arithmeticity of Case 34 (Q - rank one case). In this case




1 2 3 4
1 1 2 1 3
α = 0, , , ,
, , , ,
;
β=
5 5 5 5
2 3 3 4 4
f (x) = x5 − 1;

g(x) = x5 + 2x4 + 3x3 + 3x2 + 2x + 1.

1

24

JITENDRA BAJPAI AND SANDIP SINGH

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
5

 −7


5
Q=


5



−7

−7

5

5

5

−7

5

−7

5

−7

5

−7

5

5

5

−7

0

1



0
X =


 −1


−7 

5 


5 ;


−7 
5

0

−2

3

2 

3 


4 ,


2 

−1

−4

6

−2

−3

7

−3

−1

5

−5

−2

3

−1

1

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively
0











0

0

0

0

24

−36

12

0

−36

24

0

0

12

0

24

−12

0

0

0

−12 

0 


0 ,


0 
0

 1

 1


 0


 0
1

1

−5

5

0

−1

1

0

2

−3

0

1

−2

−1

−2

4

−1 

0 


1 ,


0 

−1

 1

 1


 0


 0
1

−3

1

3

−2

2

0

0

2

−3

0

1

−2

−1

−2

4

1 

1 


1 .


0 

−1

A computation shows that the Q - rank of the orthogonal group SOQ is one. If we denote by
c = a−1 b,

d = [a, b−1 ],

w3 = w22 (d2 w22 )−1 w22 ,

w1 = d(ede−1 )2 , w2 = c[a2 , b],
2
w5 = w2 w3−5 w2−1 w4−2 , w6 = d98 w5 ,

e = [a−4 , b−4 ],

w4 = w1−4 w33 ,

then the final three unipotent elements are
q1 = w6−432 q2294 ,

q2 = w146 w6 ,

q3 = w4432 q230 ,

which are respectively
 1

 0


 0


 0

0

0

0

−59616

1

0

0

0

1

0

0

0

1

0

0

0

 1

 0


 0


 0

740430720 

−9936 


−14904  ,


−24840 

0

1

0

−432

0

1

0

0

0

1

0

0

0

1

0

0

0

−23328 

216 


108  ,


−108 
1

 1

 0


 0


 0

1296

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

−279936 

−432 


−648  .


216 
1

3.15. Arithmeticity of Case 35 [(f(x), g(x)), (-f(-x), -g(-x)) case]. In this case




1 1 1 2 2
1 1 5 5
, , , ,
;
β=
α = 0, , , ,
6 6 6 6
2 3 3 3 3
f (x) = x5 − 3x4 + 5x3 − 5x2 + 3x − 1;

g(x) = x5 + 3x4 + 5x3 + 5x2 + 3x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
4

0



Q =  −5


0



13

0

−5

0

4

0

−5

0

4

0

−5

0

4

0

−5

0

13 

0 


−5  ;


0 
4

 −1

3



X =  −4


3

−1

−5

1/4

7/4
−19/4

13

−3/4

−18

1

21/2

14

−1/2

−13/2

−4

1/2

8

−5 

15 


−26  ,


18 
−20

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

25

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively
0

0

0

0

−36

0

1/2

0

0

−36

0

0

−18

0

0

0











0

0

0
0

 −5/2

 9/16


 −9/2


1/4


−18 

0 


0 ,


0 

15/2

 −19/16


27/2



21/4

2

5/8

−15/8

41/16

−11/64

−63/64

−45/2

−1/8

−117/8

1/4

−3/16

−7/16

−1/2

−1/8

−9/8

0

0



−23/2

57/2

−35/8

−71/16

45/64

833/64

99/2

−73/8

−1269/8

81/4

−43/16

−647/16

15/2

−9/8

−137/8

−655/8

3/2 

51/16 


81/2  ,


11/4 
7/2

403/2 

−509/16 


801/2  .


411/4 
87/2

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
d = aba−2 ,

e = aba,

t = ba−1 b,

w4 = dw3 dw3−1 ,

w1 = [a−6 , e],

w5 = w3 dw3−1 d,

w2 = a−48 w1 ,

w6 = [w5 , a6 ],

w3 = tb−6 t−1 ,

w7 = [w4 , a6 ],

then the final three unipotent elements are
q1 = w6 w7−1 w1−10 w2 w1−2 w23 ,

q3 = w2384 q1−60 ,

q2 = dq1 d,

which are respectively
 1

 0


 0


 0
0

0

0

−768

1

0

0

0

1

0

0

0

1

0

0

0

0 

384 


0 ,


0 
1

 1

 0


 0


 0
0

0

−4608

0

1

0

0

0

1

0

0

0

1

0

0

0

3.16. Arithmeticity of Case 36. In this case


1 1 5 5
;
α = 0, , , ,
6 6 6 6
f (x) = x5 − 3x4 + 5x3 − 5x2 + 3x − 1;

382205952 

0 


−165888  ,


0 
1

β=



 1

 0


 0


 0

−3072

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

1 1 2 1 3
, , , ,
2 3 3 4 4

0 

0 


0 .


1536 
1



g(x) = x5 + 2x4 + 3x3 + 3x2 + 2x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
13

2



Q =  −14


 −7


22

2

−14

−7

13

2

−14

2

13

2

−14

2

13

−7

−14

2

22 

−7 


−14  ;


2 
13

1

 −1


0
X=


1



−1

7

0

19

−8

3/49

166/7

2

1/49

242/7

5

−1/49

185/7

−6

4/49

114/7

36 

42 


64  ,


48 
26

26

JITENDRA BAJPAI AND SANDIP SINGH

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

252

0

3/49

0

0

252

0

0

−72

0

0

0

20

 −3


0



0



−72 

0 


0 ,


0 
0

2

 −2/7


0



1



−1/2

127

−73/49

−2042/7

−19

11/49

306/7

0

1

−84

0

0

20

0

0

−21/2

4 

−66/49 


−240  .


72/7 

0

19

−1/49

10/7

−19/7

1/343

−24/49

0

1

−84

6

−4/49

26/7

−3

2/49

−33/14

−464 

3392/49 


−240  ,


254/7 
−19

−6

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
d = ba6 b−1 ,

w2 = b−1 a6 b,

w1 = ba−1 ba−4 ,

w3 = [w2−1 , w1 ],

w4 = dw1 d−1 w1 ,

w5 = w4 w3−1 ,

then the final two unipotent elements, corresponding to the highest and second highest roots, are
q 1 = a6 ,

q2 = w57 q12263 ,

which are respectively
 1

 0


 0


 0
0

0

0

84

1

0

0

0

1

0

0

0

1

0

0

0

 1

 0


 0


 0

0 

24 


0 ,


0 

0

1

0

−24

0

1

0

0

0

1

0

0

0

1

0

0

0

338688 

0 


−28224  ,


0 
1

and the arithmeticity of Γ(f, g) follows from [14, Theorem 3.5] (cf. Remarks 3.2, 3.3, 3.4).
3.17. Arithmeticity of Case 37. In this case


1 1 5 5
α = 0, , , ,
;
6 6 6 6
f (x) = x5 − 3x4 + 5x3 − 5x2 + 3x − 1;

β=



1 1 1 3 3
, , , ,
2 4 4 4 4



g(x) = x5 + x4 + 2x3 + 2x2 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
73

17



Q =  −71


 −55


73

17

−71

−55

73

17

−71

17

73

17

−71

17

73

−55

−71

17

73 

−55 


−71  ;


17 
73

 2

 2


X = 2


 2
0

0

0

0
−8

0

2

−6

2

1

3

−1

−7/2

−9

5

−7/2

1 

−3 


6 ,


−4 
4

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

27

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

−432

0

144

0

0

−432

0

0

144

0

0

0

144 

0 


0 ,


0 
0

0

 −3/2


−3



−1



0

3/2

−3/2

9/4

−6

7/2

−3/2

−6

2

3

0

−1

3

−12

8

−8

0

 −3/2


−3



−1


0 

25/12 


5/2  ,


1/2 



0

4

6

−4

4

−6

7/2

−3/2

−6

2

3

0

−1

3

−3

3

−9/2

−2 

25/12 


5/2  .


1/2 
0

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
d = ba−6 b−1 ,

w1 = (ab−1 a)8 ,

e = a−1 b−4 a,

w2 = w12 e,

w3 = w2−3 w12 ,

w4 = dw2 d−1 ,

then the final three unipotent elements are
q2 = w24032 q130 ,

q1 = [w4 , w2−7 ],

q3 = w3112 q1−1 ,

which are respectively
 1

 0


 0


 0
0

0

0

−1344

1

0

0

0

1

0

0

0

1

0

0

0

0 

−448 


0 ,


0 
1

 1

 0


 0


 0
0

0

−16128

0

1

0

0

0

1

0

0

0

1

0

0

0

3.18. Arithmeticity of Case 38. In this case


1 3 5 7
;
α = 0, , , ,
8 8 8 8
f (x) = x5 − x4 + x − 1;

−130056192 

0 


16128  ,


0 
1

β=



 1

 0


 0


 0
0

1 1 1 2 2
, , , ,
2 3 3 3 3

1344

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0 

0 


0 .


448 
1



g(x) = x5 + 3x4 + 5x3 + 5x2 + 3x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
73

−53



1
Q=


55



−71

−53

1

55

73

−53

1

−53

73

−53

1

−53

73

55

1

−53

 −2

0



1
X =


2


−71 

55 


1 ;


−53 
73

−1

71/7

−4/49

9

−50/7

4/49

−2

−3

0

−7

26/7

8/49

−11

118/7

4/49

5

7 

−5 


−1  ,


4 
13

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

−252

0

144/49

0

0

−252

0

0

−36

0

0

0

−36 

0 


0 ,


0 
0

3

 −5/7


1



 29/49


1

−24/7

12/49

−13

64/49

−60/343

30/7

46/7

−37/49

−6

11/343

152/2401

−125/49

−3/7

12/49

−6

−2 

6/7 


3 ,


3/49 
0

28

JITENDRA BAJPAI AND SANDIP SINGH


−4

−2



0



−6/7


3

802/7

40/49

22

1126/49

−24/343

75/7

164/7

−33/49

−1

8389/343

436/2401

230/49

−239/7

4/49

−16

89 

123/7 


16  .


926/49 
−26

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
e = a−1 b6 a,

c = a−1 b, d = cbc,

w1 = [e, a4 ],

w2 = w149 e−48 ,

w3 = ed−3 , w4 = w3 w1−1 w3 w1 ,

then the final three unipotent elements are
q1 = q2−1 w26 ,

q2 = w47 e−247 ,

q3 = e,

which are respectively
 1

 0


 0


 0
0

0

0

−3528

1

0

0

0

1

0

0

0

1

0

0

0

0 

504 


0 ,


0 

 1

 0


 0


 0

1

0

0

144

0

1

0

0

0

1

0

0

0

1

0

0

0

3.19. Arithmeticity of Case 39. In this case


1 3 5 7
;
α = 0, , , ,
8 8 8 8
f (x) = x5 − x4 + x − 1;

127008 

0 


1764  ,


0 
1

β=



 1

 0


 0


 0

42

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

1 1 2 1 3
, , , ,
2 3 3 4 4

0 

0 


0 .


−6 
1



g(x) = x5 + 2x4 + 3x3 + 3x2 + 2x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
13

 −11


1
Q=


13



−11

−11

1

13

13

−11

1

−11

13

−11

1

−11

13

13

1

−11

−11 

13 


1 ;


−11 
13

 −8

 −8


X =  −8


2

−2

4

−3/2

−17/2

0

−3

−12

0

−3

−16

−4

3/2

1/2

0

0

0

and the matrices of Q, A and B, with respect to the new basis, are
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

−192

0

36

0

0

−192

0

0

48

0

0

0

3/2

 39/16


−5



1/2


48 

0 


0 ,


0 



0

−3

5/2

−3


 73/16


−7



 −1/2



−11

19/8

−33/8

69/64

297/64

6

−11/4

−35/4

1

3/8

7/8

10

−9/4

−21/4

3/8

19/8

−33/8

69/64

297/64

6

−11/4

−35/4

1

3/8

7/8

10

−9/4

−21/4

−3/4 

−39/32 


5/2  .


−1/4 
3/2

−1

X t QX,

3/8

−3

1 

1 


1 ,


−2 

a = X −1 AX and

−5/4 

−73/32 


7/2  ,


1/4 
11/2

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

29

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
c = a−1 b,

d = [a−4 , b3 ],

e = ab,

w1 = cdc,

w2 = ce,

w4 = w12 w3 ,

w3 = w2 w1 w2−1 ,

then the final two unipotent elements, corresponding to the highest and second highest roots, are
q2 = w4−96 q1132 ,

q1 = [w1 , w3 ],
which are respectively
 1

 0


 0


 0
0

0

0

96

1

0

0

0

1

0

0

0

1

0

0

0

 1

 0


 0


 0

0 

24 


0 ,


0 

0

1

0

−576

0

1

0

0

0

1

0

0

0

1

0

0

0

−221184 

0 


768  ,


0 
1

and the arithmeticity of Γ(f, g) follows from [14, Theorem 3.5] (cf. Remarks 3.2, 3.3, 3.4).
3.20. Arithmeticity of Case 40. In this case


1 3 5 7
α = 0, , , ,
;
8 8 8 8
f (x) = x5 − x4 + x − 1;

β=



1 1 1 3 3
, , , ,
2 4 4 4 4



g(x) = x5 + x4 + 2x3 + 2x2 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
1

 −3


1
Q=


5



1

−3

1

5

1

−3

1

−3

1

−3

1

−3

1

5

1

−3

1 

5 


1 ;


−3 
1

0

1



0
X =


 −1


0

1

8/9

1/6

1

8/9

1/6

4

8/9

−1/3

1

8/9

−17/6

1

8/9

1/6

1 

1/2 


2 ,


1/2 
0

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0
0

0

0

24

0

64/9

0

0

24

0

0

−8

0

0

0

1/2
−8  
 
0   −1/18
 
 
−1
0 ,
 
 
0   −1/3
0

1

−3/2

−4/3

5/4

5/18

32/81

−115/108

2

13/9

−5/3

−1/3

16/27

−13/18

−1

−8/9

17/6

1/2
3/4  
 
7/36   −1/18
 
 
−1
1/2  , 
 
 
−5/6   −1/3

−1/2

1/2

1

4/9

19/12

1/6

8/27

−13/12

0

−1/3

−2

−1

0

−5/6

−1

−8/9

17/6

3/4 

7/36 


1/2  .


−5/6 

−1/2

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
w1 = [a, b−1 ],

w2 = [a−4 , b3 ],

w3 = ba4 b−1 ,

w4 = [w3 , w1 ],

w5 = w2 w1 w2−1 w1−89 ,

then the final three unipotent elements are
q1 = w53 q2−8 ,

q2 = w5 w4−10 ,

q3 = (w118 q2 )40 q1 ,

30

JITENDRA BAJPAI AND SANDIP SINGH

which are respectively
 1

 0


 0


 0
0

0

0

240

1

0

0

0

1

0

0

0

1

0

0

0

0 

80 


0 ,


0 
1

 1

 0


 0


 0
0

0

32

0

1

0

0

0

1

0

0

0

1

0

0

0

 1

 0


 0


 0

576 

0 


36  ,


0 

−1440

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

1

0 

0 


0 .


−480 
1

3.21. Arithmeticity of Case 41 (Q - rank one case). In this case




1 1 2 1 3
1 5 7 11
, , , ,
;
β=
α = 0, , , ,
12 12 12 12
2 3 3 4 4
f (x) = x5 − x4 − x3 + x2 + x − 1;

g(x) = x5 + 2x4 + 3x3 + 3x2 + 2x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
5

 −5


3
Q=


1



−5

3

1

5

−5

3

−5

5

−5

3

−5

5

1

3

−5

−1

−1 

1 


3 ;


−5 
5

 0

 0


X= 1


 1
0

5

−3

2

−4

9

1

−5

13

0

−4

10

2

−2

1

3

2 

1 


1 ,


−3 

−3

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively
0











0

0

0

0

24

−12

−36

0

−12

−24

60

0

−36

60

24

12

0

0

0

12  
 
0  
 
 
0 ,
 
 
0  
0

1/6

19/6

−5/2

8/3

0

0

−4

1

0

0

−1

0

1/6

−3/2

−5/6

8/3

−1/6

1/2

−13/6

1/3

−49/6  
 
0  
 
 
1 ,
 
 
−19/6  
−5/6

1/6

3/2

7/6

−10/3

0

0

−4

1

0

0

−1

0

1/6

1/2

−11/6

−1/3

−1/6

1/2

−13/6

1/3

−13/6 

0 


1 .


−1/6 
−5/6

A computation shows that the Q - rank of the orthogonal group SOQ is one. If we denote by
c = a−1 b,

d = [a−1 , b−5 ],

w2 = c[d, w1 ]c,

e = bab−1 ,

w3 = [e, r −1 ],

r = [b−1 , a],

w4 = w2 w3 w2 ,

w7 = [w4 , r −1 ]w539 ,
then the final three unipotent elements are
q1 = w6504 w7306 ,

s = b−1 ab,

w5 = [s, r −1 ],

w1 = [a−1 , b2 ],

w6 = w4 w5−1 w4 w54 ,

w8 = [w1 , r −1 ],

q2 = w6360 w7342 ,

q3 = w862208 q2−78 ,

which are respectively
 1

 0


 0


 0

0

0

0

62208

1

0

0

0

1

0

0

0

1

0

0

0

−806215680  
 
−57024  
 
 
−36288  , 
 
 
−25920  
1

1

0

62208

0

0

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

−806215680  
 
−67392  
 
 
−25920  , 
 
 
−36288  
1

1

−1866240

0

0

0

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

−4208445849600 

4510080 


2021760  .


1710720 
1

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

3.22. Arithmeticity of Case 42. In this case


1 5 7 11
α = 0, , , ,
;
12 12 12 12
f (x) = x5 − x4 − x3 + x2 + x − 1;

β=



1 1 2 3 4
, , , ,
2 5 5 5 5

31



g(x) = x5 + 2x4 + 2x3 + 2x2 + 2x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
21

 −19


11
Q=


1



−19

11

1

21

−19

11

−19

21

−19

11

−19

21

1

11

−19

−9

 −3

 −4


X =  −2


5


−9 

1 


11  ;


−19 
21

−2

−5/4

7/4

−1

0

3

1

5/2

7/2

2

15/4

7/4

0

0

0

4

−15/4 

−4 


−3/2  ,


1/4 
−1

and the matrices of Q, A and B, with respect to the new basis, are X t QX, a = X −1 AX and
b = X −1 BX; which are respectively










0

0

0

0

0

0

0

−10

0

0

25

0

0

−10

0

0

−40

0

0

0

33/8

 −25/2


−3



9



−40 

0 


0 ,


0 

3/8

 −11/2


−1



1


17/16

45/32

9/32

−9/4

−5/8

15/8

−11/4

−3/4

3

15/4

3/4

9/4

15/8

−5/8

9/32

−9/4

−5/8

15/8

−3/2

−11/4

−3/4

3

15/4

3/4

9/4

15/8

−5/8

9/2

45/32

−3/2

15/2

0



17/16

3/32 

−11/8 


−1/4  ,


1/4 
9/8

33/32 

−25/8 


−3/4  .


9/4 
15/8

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
c = a−1 b,

d = b−4 a6 b4 ,

w4 = w1−384 w3 , w5 = w296 w4 ,

e = da4 da−4 ,

w1 = [e, a6 ],

w2 = [w1 , a3 ],

w6 = cw3 c, w7 = da6 d−1 a6 ,

w3 = [w1−24 , w2 ],

w8 = w65 w7 w3−1 w7−1 , w9 = w8−1 w5 w8 ,

then the final two unipotent elements, corresponding to the highest and second highest roots, are
q1 = w3 ,

q2 = w93840 q1238886400 ,

which are respectively
 1

 0


 0


 0
0

0

0

−3840

1

0

0

0

1

0

0

0

1

0

0

0

0 

15360 


0 ,


0 
1

 1

 0


 0


 0
0

0

7372800

0

1

0

0

0

1

0

0

0

1

0

0

0

43486543872000 

0 


11796480  ,


0 
1

and the arithmeticity of Γ(f, g) follows from [14, Theorem 3.5] (cf. Remarks 3.2, 3.3, 3.4).

32

JITENDRA BAJPAI AND SANDIP SINGH

3.23. Arithmeticity of Case 43. In this case


1 5 7 11
;
α = 0, , , ,
12 12 12 12

β=

f (x) = x5 − x4 − x3 + x2 + x − 1;



1 1 3 5 7
, , , ,
2 8 8 8 8



g(x) = x5 + x4 + x + 1.

The matrix (up to scalar multiplication) of the quadratic form Q, with respect to the standard
basis {e1 , e2 , e3 , e4 , e5 }, and the change of basis matrix X, are
1

 −3


1
Q=


1



1

−3

1

1

1

−3

1

−3

1

−3

1

−3

1

1

1

−3

0

1



0
X =


 −1

1 

1 


1 ;


−3 

1



−1

−3/4

1/6

−3/4

3/2

0

1

0

1

3/4

−1/6

and the matrices of Q, A and B, with respect to the new basis, are
b = X −1 BX; which are respectively
 0

 0


 0


 0
4

0

0

0

0

0

16/3

0

1

0

16/3

0

0

0

0

0

4 

0 


0 ,


0 
0

1

−1/8



3



3/4



1

−1/4

−1/6

1/8

11/32

−49/48

1

−9/4

7/6

−3/4

−9/16

1/8

0

1

0

−1

0 

9/8 


−3  ,


−3/4 
2

2 

−1 


−2  ,


2 

−1/6

−1

0

1

3/4

1

−1/8



3



3/4



3

X t QX,

a = X −1 AX and

−1

−7/4

1/6

1/4

7/16

−25/24

−2

−9/2

5/3

−3/2

−9/8

1/4

0

1

0

−1

−6 

3/2 


−12  .


−3 
2

It is clear from the above computation that the Q - rank of the orthogonal group SOQ is two. If
we denote by
w1 = [a−1 , b], w2 = a3 b4 a−3 , w3 = (a3 b4 )4 , w4 = w2 w1 w2 ,
then the final three unipotent elements are
q1 = q22 w424 w3−6 ,

q2 = w112 w3−3 ,

q3 = (w312 q2 )−8 q1 ,

which are respectively
 1

 0


 0


 0
0

0

0

−64

1

0

0

0

1

0

0

0

1

0

0

0

0 

48 


0 ,


0 
1

 1

 0


 0


 0
0

0

12

0

1

0

0

0

1

0

0

0

1

0

0

0

−288 

0 


−48  ,


0 
1

 1

 0


 0


 0
0

−384

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0 

0 


0 .


288 
1

References
[1] F. Beukers, G. Heckman, Monodromy for the hypergeometric function n Fn−1 , Invent. math. 95 (1989), no. 2,
325-354.
[2] C. Brav, H. Thomas, Thin Monodromy in Sp(4), Compositio Math. 150 (2014), no. 3, 333-343.
[3] J. Hofmann, D. van Straten, Some monodromy groups of finite index in Sp4 (Z), J. Aust. Math. Soc. 99(2015),
no. 1, 48-62.
[4] E. Fuchs, The ubiquity of thin groups. Thin groups and superstrong approximation, 73-92, Math. Sci. Res. Inst.
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[5] E. Fuchs, C. Meiri, P. Sarnak, Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 8, 1617-1671.

ON ORTHOGONAL HYPERGEOMETRIC GROUPS OF DEGREE FIVE

33

[6] A. H. M. Levelt, Hypergeometric functions, Doctoral thesis, University of Amsterdam, 1961.
[7] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag Berlin Heidelberg New York (1972).
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no. 2, 365-373.
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Math. Res. Not. (IMRN), 2015 (2015), no. 18, 8874-8889.
[11] ———–, Orthogonal hypergeometric groups with a maximally unipotent monodromy, Exp. Math. 24 (2015),
no. 4, 449-459.
[12] ———–, Arithmeticity of some hypergeometric monodromy groups in Sp(4), J. Algebra 473 (2017), 142-165.
[13] S. Singh, T. N. Venkataramana, Arithmeticity of certain symplectic hypergeometric groups, Duke Math. J. 163
(2014), no. 3, 591-617.
[14] T. N. Venkataramana, Zariski dense subgroups of arithmetic groups, J. Algebra 108 (1987), no. 2, 325-339.
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Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
E-mail address: jitendra@mpim-bonn.mpg.de, singhs@mpim-bonn.mpg.de

