On Orthogonal Hypergeometric Groups of Degree Five
Jitendra Bajpai, Sandip Singh
TL;DR
This paper classifies degree five hypergeometric groups with roots of unity, identifying which have finite, rank one, or rank two Zariski closures, and determines the arithmeticity of many rank two cases.
Contribution
It provides a comprehensive classification of orthogonal hypergeometric groups of degree five, including new results on the arithmeticity of rank two cases.
Findings
77 possible polynomial pairs with roots of unity
4 finite monodromy groups identified
17 groups with rank one Zariski closure
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 closures have 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…
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On Orthogonal Hypergeometric Groups of Degree Five
Jitendra Bajpai and Sandip Singh
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Mathematisches Institut, Georg-August Universität Göttingen, Germany [email protected]
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India
Abstract.
A computation shows that there are (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 of these pairs correspond to finite monodromy groups and only pairs correspond to monodromy groups, for which, the Zariski closures have real rank one.
There are remaining pairs, for which, the Zariski closures of the associated monodromy groups have real rank two. It follows from Venkataramana [16] that of these pairs correspond to arithmetic monodromy groups and the arithmeticity of other cases follows from Singh [11]. In this article, we show that of the remaining rank two cases correspond to arithmetic groups.
Key words and phrases:
Hypergeometric group, Monodromy representation, Orthogonal group
2010 Mathematics Subject Classification:
Primary: 22E40; Secondary: 32S40; 33C80
1. Introduction
We consider the monodromy action of the fundamental group of on the solution space of the type hypergeometric differential equation
[TABLE]
on , having regular singularities at the points , where the differential operator is defined as the following:
[TABLE]
where , and , .
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 , such that , for all , 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 generated by the companion matrices and of the polynomials
[TABLE]
respectively, and the monodromy is defined by , , , where are, respectively, the loops around , which generate the of modulo the relation .
We now denote the hypergeometric group by which is a subgroup of generated by the companion matrices of the polynomials . We consider the cases where the coefficients of are integers with , (for example, one may take as product of cyclotomic polynomials); in these cases, . We also assume that form a primitive pair [1, Definition 5.1], are self-reciprocal and do not have any common root.
Beukers and Heckman [1, Theorem 6.5] have completely determined the Zariski closures of the hypergeometric groups which is briefly summarized as follows:
- •
If is even and , then the hypergeometric group preserves a non-degenerate integral symplectic form on and is Zariski dense, that is, .
- •
If is infinite and , then preserves a non-degenerate integral quadratic form on and is Zariski dense, that is, .
- •
It follows from [1, Corollary 4.7] that is finite if and only if either or ; and in this case, we say that the roots of and interlace on the unit circle.
We call a hypergeometric group arithmetic, if it is of finite index in ; and thin, if it has infinite index in [9].
The following question of Sarnak [9] has drawn attention of many people: determine the pairs of polynomials , for which, the associated hypergeometric group is arithmetic. There have been some progress to answer this question.
For the symplectic cases: infinitely many arithmetic in (for any even ) are given by Singh and Venkataramana in [13]; some other examples of arithmetic in are given by Singh in [10] and [12]; examples of thin in are given by Brav and Thomas in [2]; and in [3], Hofmann and van Straten have determined the index of in for some of the arithmetic cases of [10] and [13].
For the orthogonal cases: when the quadratic form has signature , Fuchs, Meiri and Sarnak give (infinite) families (depending on odd and ) of thin in [5]; when the quadratic form has signature with , (infinite) families of arithmetic are given by Venkataramana in [16]; an example of thin in is given by Fuchs in [4]; and examples of arithmetic in are given by Singh in [11], which deals with the orthogonal hypergeometric groups of degree five with a maximally unipotent monodromy, and these cases were inspired by the symplectic hypergeometric groups associated to Calabi-Yau threefolds [10].
This article resulted from our effort to first write down all possible pairs (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: and ; and then to answer the question to determine the arithmeticity or thinness of the associated hypergeometric groups . Note that, these pairs of the polynomials satisfy the conditions of Beukers and Heckman [1] and hence the associated hypergeometric group is either finite or preserves a non-degenerate integral quadratic form on , and is Zariski dense in the orthogonal group of the quadratic form .
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 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 .
We note that the roots of the polynomials and interlace for the pairs of Table 5 and hence the hypergeometric groups associated to these pairs are finite [1, Corollary 4.7]. The quadratic forms associated to the pairs of Tables 1 and 7 have signature , and the thinness of associated to the pairs of Table 1 follows from Fuchs, Meiri, and Sarnak [5]. The arithmeticity or thinness of associated to the other pairs of Table 7 is still unknown.
We also note that the quadratic forms associated to the remaining pairs (cf. Tables 2, 3, 4, 6) have the signature and the arithmeticity of associated to the pairs of Table 2 follows from Venkataramana [16]; and the arithmeticity of associated to the other pairs of Table 3 follows from Singh [11]. Therefore, before this article arithmeticity of only hypergeometric groups was known and the arithmeticity or thinness of the remaining hypergeometric groups was unknown.
In this article, we show the arithmeticity of more than half of the remaining hypergeometric groups . In fact, we obtain the following theorem:
Theorem 2**.**
The hypergeometric groups associated to the pairs of polynomials in Table 4 are arithmetic.
Therefore, of the remaining pairs correspond to arithmetic hypergeometric groups; and the arithmeticity or thinness of the hypergeometric groups associated to the remaining 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 hospitalities. We thank the Mathematisches Forschungsinstitut Oberwolfach where we met T.N. Venkataramana, and thank him for many discussions on the subject of the paper. We thank Wadim Zudilin for discussions at MPI. We also thank the referee for valuable comments and suggestions.
JB would like to extend his thanks to the Mathematisches Institut, Georg-August Universität Göttingen for their support and hospitality during the final revision of the article. JB’s work was financially supported by ERC Consolidator grant 648329 (GRANT).
SS gratefully acknowledges the support of the INSPIRE Fellowship from the Department of Science and Technology, India.
2. Tables
In this section, we list all possible (up to scalar shifts) pairs of degree five polynomials, which are products of cyclotomic polynomials, and for which, the pair form a primitive pair [1], and , so that the associated monodromy group preserves a quadratic form , and as a Zariski dense subgroup (except for the cases of Table 5). Note that once we know the parameters , the polynomials can be determined by using the Equation (1.2).
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 (up to scalar multiplications) preserved by the hypergeometric groups of Theorem 2, and then show that the real rank of the orthogonal group is two, and the - rank is either one or two. We then form a basis of , which satisfy the following condition: in - rank two cases, the matrix form of the quadratic form , with respect to the basis , is anti-diagonal; and in - rank one cases, the vectors are - isotropic non-orthogonal vectors (that is, and ), and the vectors are - orthogonal to the vectors .
Let be the parabolic - subgroup of , which preserves the following flag:
[TABLE]
and be the unipotent radical of . It can be checked easily that the unipotent radical is isomorphic to (as a group), and in particular, is a free abelian group isomorphic to .
We prove Theorem 2 (except for the cases 4, 4, and 4 of Table 4) by using the following criterion of Raghunathan [8], and Venkataramana [15] (cf. [16, Theorem 6]): If is Zariski dense and intersecting in a finite index subgroup of , then has finite index in . Note that, the criterion of Venkataramana [15] (for - rank one) and Raghunathan [8] (for - rank two) are for all absolutely almost simple linear algebraic groups defined over a number field , and we have stated in a way to use it to prove our theorem.
To prove Theorem 2 for the cases 4, 4 and 4 of Table 4 (note that, for all these cases, the corresponding orthogonal groups have - rank two), we use the following criterion of Venkataramana [14, Theorem 3.5]: If is a Zariski dense subgroup and intersecting the highest and second highest root groups of non-trivially, then has finite index in .
For an easy reference to the root system, and the structures of the corresponding unipotent subgroups of (of - rank two), we refer the reader to Remark 3.2 (cf. [11, Section 2.2]).
Notation
Let be a pair of integer coefficient monic polynomials of degree , which have roots of unity as their roots, do not have any common root, form a primitive pair, and satisfy the condition: and . Let be the companion matrices of respectively, and be the group generated by and . Let , be the identity matrix; be the standard basis vectors of over ; and be the last column vector of , that is, . Let be the non-degenerate quadratic form on , preserved by . Note that, is unique only up to scalars, and its existence follows from Beukers and Heckman [1].
It is clear that (since ), and hence is - orthogonal to the vectors (since for ); and (since is non-degenerate). We may now assume that .
It can be checked easily that the set (similarly ) forms a basis of (cf. [11, Lemma 2.1]). Therefore, to determine the quadratic form on , it is enough to compute , for . Also, since is - orthogonal to the vectors , and (say), we get
[TABLE]
We compute the signature of the quadratic form by using [1, Theorem 4.5], which says the following: By renumbering, we may assume that and . Let for . Then, the signature of the quadratic form is determined by the equation
[TABLE]
Now using the above equality with the non-degeneracy of the quadratic form (that is, ), one can check easily that all the quadratic forms associated to the pairs of polynomials in Tables 4 and 6 have signature , that is, the associated orthogonal groups have real rank two. Therefore, the quadratic forms in the cases of Tables 4 and 6, have two linearly independent isotropic vectors in (by the Hasse-Minkowski theorem), which are not - orthogonal, that is, the associated orthogonal groups have - rank .
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 the change of basis matrix (cf. second paragraph of Section 3), that is,
[TABLE]
We also denote by the matrices respectively, with respect to the new basis, that is,
[TABLE]
We compute the quadratic forms on by using Equation (3.1), and denote by only, the matrix forms of the quadratic forms , with respect to the standard basis of , so that the conditions (note that , denote the transpose of the respective matrices)
[TABLE]
are satisfied.
Let be the parabolic - subgroup of (defined in the third paragraph of Section 3), and let be the unipotent radical of . Since is a free abelian group isomorphic to , to show that has finite index in , it is enough to show that it contains three linearly independent vectors in . In the proof below, we denote the three corresponding unipotent elements by , , and (note that these are the words in and which are respectively some conjugates of and (see the last paragraph), and hence (with respect to some basis of ) ); and in some of the - rank two cases, we are able to show only two unipotent elements and inside , which correspond, respectively, to the highest and second highest roots (Remarks 3.2, 3.3, 3.4), and the arithmeticity of 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 . Note that in the case of symplectic hypergeometric groups (when is even and ) the matrix itself is a non-trivial unipotent element, and to produce more unipotent elements in , one can take the conjugates of 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 is not unipotent (since one of the eigen value of is and all other eigen values are ) and if we take (to get rid of the eigen value ), it becomes the identity element. Therefore, to find a non-trivial unipotent element in , we take conjugates of by the elements of , 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 - 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
[TABLE]
inside the orthogonal group, and if is the character of defined by
[TABLE]
then the roots of the orthogonal group are given by If we fix a set of simple roots then the set of positive roots , the set of negative roots , and , are respectively the highest and second highest roots in .
The unipotent subgroups corresponding to the highest and the second highest roots are respectively given by
[TABLE]
where , and , and is a basis of , with respect to which, the matrix associated to the quadratic form has the anti-diagonal form.
Remark 3.3**.**
Since is Zariski dense in the orthogonal group (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 have - rank two, it is enough to show that
[TABLE]
are finite. Since
[TABLE]
and
[TABLE]
the arithmeticity of the hypergeometric groups in Table 4, for which, the associated orthogonal groups have - rank two, also follows from [14, Theorem 3.5].
Remark 3.4**.**
In the three cases 4, 4 and 4, we find the unipotent elements (in ) 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 cases of Table 4, so we do not put much effort in computing the third unipotent element (to show that ). Also, once we find that the group has finite index in the integral orthogonal group (using [14, Theorem 3.5]) it follows trivially that contains the unipotent elements corresponding to each roots, and hence is also finite.
Remark 3.5**.**
In some of the remaining cases (cf. Table 6), for which, the associated quadratic forms have signature , 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.
We now give a detailed explanation of the computation for finding out the unipotent elements inside the hypergeometric group in the first case (Case 4) 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 4
In this case
[TABLE]
[TABLE]
Let and be the companion matrices of and respectively, and let . Then
[TABLE]
Let be the standard basis of over , and let . Then , , , ; and hence by the Equation (3.1) it follows that, with respect to the basis , the matrix of the quadratic form (up to scalar multiplication) preserved by the hypergeometric group is given by
[TABLE]
We now let be the change of basis matrix from the basis to the standard basis . Then, a computation shows that
[TABLE]
and, with respect to the standard basis , the matrix (up to scalar multiplication) of the quadratic form preserved by the hypergeometric group is given by
[TABLE]
(here we multiply by , just to get rid of the denominators) where denotes the transpose of the matrix . Thus, we find that
[TABLE]
After computing the isotropic vectors of the quadratic form , we change the standard basis to the basis , for which, the change of basis matrix is given by
[TABLE]
With respect to the new basis, the matrices of , and are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then
[TABLE]
[TABLE]
and the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
It is clear from the above computation that the group generated by is a subgroup of , and has finite index in , the integer points of the unipotent radical of the parabolic subgroup , which preserves the flag
[TABLE]
Therefore is of finite index in . Since is also Zariski dense in the orthogonal group (by Beukers and Heckman [1]), the arithmeticity of 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 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 (say) in by which is .
3.2. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then
[TABLE]
and the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.3. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then
[TABLE]
[TABLE]
[TABLE]
and the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.4. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.5. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.6. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.7. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.8. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.9. Arithmeticity of Case 4 ( - rank one case)
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
A computation shows that the - rank of the orthogonal group is one. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.10. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.11. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.12. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.13. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.14. Arithmeticity of Case 4 ( - rank one case)
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
A computation shows that the - rank of the orthogonal group is one. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.15. Arithmeticity of Case 4 [(f(x), g(x)), (-f(-x), -g(-x)) case]
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.16. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then the final two unipotent elements, corresponding to the highest and second highest roots, are
[TABLE]
which are respectively
[TABLE]
and the arithmeticity of follows from [14, Theorem 3.5] (cf. Remarks 3.2, 3.3, 3.4).
3.17. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.18. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.19. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then the final two unipotent elements, corresponding to the highest and second highest roots, are
[TABLE]
which are respectively
[TABLE]
and the arithmeticity of follows from [14, Theorem 3.5] (cf. Remarks 3.2, 3.3, 3.4).
3.20. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.21. Arithmeticity of Case 4 ( - rank one case)
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
A computation shows that the - rank of the orthogonal group is one. If we denote by
[TABLE]
[TABLE]
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
3.22. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
[TABLE]
then the final two unipotent elements, corresponding to the highest and second highest roots, are
[TABLE]
which are respectively
[TABLE]
and the arithmeticity of follows from [14, Theorem 3.5] (cf. Remarks 3.2, 3.3, 3.4).
3.23. Arithmeticity of Case 4
In this case
[TABLE]
[TABLE]
The matrix (up to scalar multiplication) of the quadratic form , with respect to the standard basis , and the change of basis matrix , are
[TABLE]
and the matrices of , and , with respect to the new basis, are , and ; which are respectively
[TABLE]
It is clear from the above computation that the - rank of the orthogonal group is two. If we denote by
[TABLE]
then the final three unipotent elements are
[TABLE]
which are respectively
[TABLE]
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] C. Brav, H. Thomas, Thin Monodromy in Sp ( 4 ) Sp 4 \mathrm{Sp}(4) , Compositio Math. 150 (2014), no. 3, 333-343.
- 3[3] J. Hofmann, D. van Straten, Some monodromy groups of finite index in Sp 4 ( ℤ ) subscript Sp 4 ℤ \mathrm{Sp}_{4}(\mathbb{Z}) , J. Aust. Math. Soc. 99 (2015), no. 1, 48-62.
- 4[4] E. Fuchs, The ubiquity of thin groups. Thin groups and superstrong approximation, 73-92, Math. Sci. Res. Inst. Publ., 61 , Cambridge Univ. Press, Cambridge, 2014.
- 5[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.
- 6[6] A. H. M. Levelt, Hypergeometric functions, Doctoral thesis, University of Amsterdam, 1961.
- 7[7] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag Berlin Heidelberg New York (1972).
- 8[8] ———–, A note on generators for arithmetic subgroups of algebraic groups, Pacific Journal of Math., 152 (1992), no. 2, 365-373.