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Goursat rigid local systems of rank four
 Danylo Radchenko^{1}Email authorView ORCID ID profile and
 Fernando Rodriguez Villegas^{2}
https://doi.org/10.1007/s406870180156y
© The Author(s) 2018
 Received: 6 March 2018
 Accepted: 17 August 2018
 Published: 5 September 2018
Abstract
We study the general properties of certain rank 4 rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form H when it exists. By a computer search, we find what we expect are all irreducible such systems all whose solutions are algebraic functions and give several explicit examples defined over \({\mathbb {Q}}\). We also exhibit one example with infinite monodromy as arising from a family of genus two curves.
1 Introduction
The question of when linear differential equations in a variable t have all of their solutions algebraic functions of t goes back to the early 1800s. In his 1897 thesis, written under the supervision of P. Painlevé, Boulanger [4] mentions a paper by J. Liouville of 1833 [17] as a possible first work on the matter. The introduction of Boulanger’s thesis offers a lucid description of the history of the question up to the time of his writing. For more recent work on the problem, see [26].
Schwarz [21] famously described all cases of algebraic solutions to the hypergeometric equation satisfied by Gauss’s series \({}_2F_1\). This was much later extended to hypergeometric equations of all orders by Beukers and Heckman [2]. In what follows, we will often refer to the better known hypergeometric local systems for comparison with [2] as our main source. For general background on local systems, monodromy representations and differential equations, see [6].
From a broader point of view, we may say that differential equations with all solutions algebraic are a special case of motivic local systems. Without attempting a rigorous definition of what this means, we will just say that such systems should be geometric in nature. Simpson conjectures in [25, p. 9] that all rigid local systems (see Sect. 2) satisfying some natural conditions are motivic. This is known for rigid local systems on \({\mathbb {P}}^1\) by the work of Katz [16], who gave a general algorithm (using middle convolution) for their construction. See also [8] for systems over a higher dimensional base and [29] for more on differential equations and arithmetic.
Goursat in his remarkable 1886 paper [10] discusses differential equations which, in later terminology, have no accessory parameters; i.e., where the local data uniquely determines the global monodromy representation. In this note, we consider his case II of rank 4 (denoted henceforth by GII). These are order four linear differential equations in a variable t with three regular singular points and semisimple local monodromies with eigenvalues of multiplicities \(2 1^2,2^2,1^4\), respectively (see Sect. 7). We will fix the singular points to be \(0,1,\infty \). In modern language, GII corresponds to certain rigid local systems of rank 4. We will always assume the local monodromies are of finite order. This a necessary condition for a local system to be motivic [19, Thm. 9], the main focus of this paper. (Note that since we assume the local monodromies are semisimple finite order is equivalent to quasiunipotent.)
In this paper, we study the general properties of GII systems; for example, we analyze when they are irreducible and describe a Hermitian form H invariant under the monodromy group when it exists. This is done in Sect. 4. As in [2], H is a key tool to understand when this group is finite. Indeed, a necessary condition is that H be definite in every complex embedding of the field of definition. Finiteness of the monodromy group is equivalent to solutions to the linear differential equations being algebraic.
We also show explicitly in Sect. 4.3 that the monodromy group can be defined in an integral way in terms of the eigenvalues of the local monodromies (the defining data). This abstractly follows from the fact that rigid systems over \({\mathbb {P}}^1\) are motivic (see [25, p. 9]); on the other hand, our construction is explicit. We find (see Sect. 3) that there is a nontrivial obstruction for the field of definition of the monodromy group. It may not be possible to define the monodromy group in its field of moduli (the field of coefficients of the characteristic polynomials of the local monodromies). This is in contrast with the hypergeometric case, for example, where by a theorem of Levelt [2, Prop. 3.3] such an obstruction does not occur. For the GII systems, the obstruction is given by a quaternion algebra over the field of moduli (see the end of Sect. 4.3 for the general case and Sect. 8 for the case where the field of moduli is \({\mathbb {Q}}\)).
As in the hypergeometric case, there are infinitely many cases of finite monodromy GII local systems which come in families. These families depend linearly on a rational parameter. For GII, there are two such families (see Sect. 12). All of these cases have imprimitive monodromy groups.
By a computer search, we find in Sect. 5 what we expect are all irreducible GII equations whose solutions are algebraic functions and give several explicit examples defined over \({\mathbb {Q}}\) in Sect. 9. In Sect. 6, we show how some GII cases can be constructed starting from a rank 4 Coxeter group by appropriate choices of pairs of commuting reflections. We exhibit in Sect. 11 one example with infinite monodromy as arising from a family of genus two curves.
We should point out that GII is a special case of rigid local systems with at least one regular semisimple local monodromy. These were classified by Simpson in [24]. Except for a sporadic case in rank 6 they consist of the hypergeometric cases and one other case in each rank \(\ge 2\). An explicit construction of the corresponding differential equations for these was given by [9]; see also [12] and [13].
We present in this paper our results with few detailed proofs, which will appear in a subsequent work. We used MAGMA [3] and PARIGP [27] for most of the calculations.
2 Rigid local systems
Following the setup and notation of [14], we consider the character variety \({\mathcal M}_\mu \) where \(\mu \) is an ordered ktuple of partitions of a positive integer n. This variety parametrizes representations of \(\pi _1(\Sigma \setminus S,*)\) to \({\text {GL}}_n({\mathbb {C}})\) mapping a small oriented loop around \(s\in S\) to a semisimple conjugacy class \(C_s\) whose generic eigenvalues have multiplicities \(\mu ^s=(\mu ^s_1,\mu ^s_2,\ldots )\), a corresponding partition in \(\mu \). Unless otherwise stated in what follows conjugation will always refer to conjugation by \({\text {GL}}_n({\mathbb {C}})\).
Goursat in his remarkable 1886 paper [10] discusses when the local monodromy data uniquely determines the representation, or in terms of the differential equation and in later terminology, when are there no accessory parameters. We want local conditions that guarantee the following. Given two ktuples of matrices \(T_s\in C_s\) and \(T'_s\in C_s\) for \(s\in S\) satisfying (1), there exists a single \(U\in {\text {GL}}_n({\mathbb {C}})\) such that \(T_s'=UT_sU^{1}\) for all \(s\in S\). The corresponding local systems (determined by the local solutions to the linear differential equation) are known as rigid local systems [16].
To have a rigid local system is to say that \({\mathcal M}_\mu \) consists of a single point. Therefore, it is necessary that the expected dimension \(d_\mu \) be zero. This is precisely Goursat’s condition [10, (5) p.113] (he only considers the case of \(g=0\)) as well as Katz’s [16], which follows from cohomological considerations.
We assume from now on that \(g=0\) and then to avoid trivial cases that \(k\ge 3\). Indeed, for \(g=0,k=1\), the group \(\pi _1(\Sigma \setminus S,*)\) is trivial and for \(g=0,k=2\) it is isomorphic to \({\mathbb {Z}}\). Note, as Goursat points out, that adding an extra puncture to S with associated partition (n) does not change the value of \(d_\mu \). Such points correspond to apparent singularities in the differential equation and may hence be safely ignored. We will assume then that the partitions \(\mu _s\) have at least two parts.
Goursat shows that with the given assumptions \(k\le n+1\) [10, top p.114] and hence there are only finitely many solutions of \(d_\mu =0\) for fixed n. He lists [10, p. 115] the cases of \(d_\mu =0\) for \(n=3\) and \(n=4\) (see below).
It turns out, however, that the condition \(d_\mu =0\) is not sufficient as the variety \({\mathcal M}_\mu \) might be empty. CrawleyBoevey [5] proved that a necessary and sufficient condition for \({\mathcal M}_\mu \) to be a point, in the case of generic eigenvalues we are considering, is that \(\mu \) corresponds to a real root of the associated Kac–Moody algebra. Without getting too deeply into the details of this condition, we present an algorithm that will allow us to determine when \({\mathcal M}_\mu \) is a point. This algorithm ultimately corresponds to Katz’s middle convolution and is simply an explicit implementation of CrawleyBoevey’s criterion. The reader may consult [18] as a general reference for this topic.

A: Replace the value n at the central node bywhere \(n_i\) are the values at the nodes closest to the central node.$$\begin{aligned} \sum _i n_i n, \end{aligned}$$

B: Shrink to a point any segment whose endpoints values are the same.

C: For each leg put new values on the nodes (not including the central node) so that the set of differences of consecutive values remains the same but appear in nondecreasing order as one moves away from the central node along the leg (so that they correspond to a partition of the value at the central node).
For our running example \(\mu =(2 1^2,2^2,1^4)\), the algorithm works as follows.
We should note that Goursat himself showed using classical tools that his case IV did not correspond to a differential equation without accessory parameters [10, p. 120] (...on devra exclure la quatrième).
3 Field of definition and field of moduli
Given a rigid local system with conjugacy classes \(C_s\) for \(s\in S\) as in Sect. 2, let \(q_s(T)\) be the characteristic polynomial of any element of \(C_s\). Let K be the field obtained by adjoining to \({\mathbb {Q}}\) the coefficients of all \(q_s\). We call K the field of moduli or simply the trace field of the local system (see below for a justification for this name). It is the smallest field F over which local monodromies \(T_s\in {\text {GL}}_n(F)\) of the required kind, i.e., \(T_s\in C_s\), may exist. But as is typical in such problems it does not mean that we can actually choose \(F=K\).
Given a collection of local monodromies giving rise to our local system, we call its field of definition the smallest extension F of \({\mathbb {Q}}\) containing all of their entries. We necessarily have \(K\subseteq F\). Note that by Levelt’s theorem [2, Prop. 3.3], in the hypergeometric case, we can always take \(F=K\), but this is not the case for Goursat’s case II that we analyze here (see Sect. 4.3).
The following is a consequence of a standard result in Galois cohomology (see [23, Chap.10, §5]); we leave the details to the reader.
Proposition 1
There exists a solution to (1) over K if and only if \(\xi \) is trivial.
Note that (2) implies that the trace of any product of \(T_s\)’s is in the trace field K. That is, K is indeed the smallest extension of \({\mathbb {Q}}\) containing the traces of all \(T\in \Gamma \).
4 Explicit solution for the Goursat case II
4.1 Criterion for irreducibility
4.2 Invariant Hermitian form
On the other hand, if the eigenvalues of \(T_s\) are invariant under the map \(z\mapsto {\bar{z}}^{1}\) for all s then the \((T_s^*)^{1}\) give another solution to (1). If our system is rigid, then there exists H satisfying (6). Up to a possible scalar factor H is a Hermitian form invariant under the monodromy group.
The set \({\mathbb {T}}^{{\text {irr}}}\) has finitely many connected components. The signature of H is constant on these components as it is continuous with integer values. We may further break the symmetry and choose the exponents satisfying \(\alpha _1<\alpha _2\) and \(\gamma _1<\gamma _2<\gamma _3<\gamma _4\) (recall that \(\exp (2\pi i \alpha _j)=a_j\) and so on). Then we find that there is a unique connected component where H is positive definite.
It is worth noting that (6) is a system of linear equations in the entries of H and can be easily solved. More generally, if \(\{A_k\}\) and \(\{B_k\}\) are two collections of matrices, then we can easily test if they are simultaneously conjugate by solving the system \(A_kXXB_k=0\). In our computations with monodromy groups, we often rely on this observation.
Proposition 2
4.3 Integrality
The matrices given by Goursat (3), when expressed in terms of the eigenvalues, have nontrivial denominators. On the other hand, as discussed in the introduction, we should be able to exhibit the monodromy group integrally. In particular, we should be able to find integral form of our local monodromies. Integrality is crucial to analyze the cases with finite monodromy (Sect. 5).
In fact, the local monodromies are definable over the ring \(R[a_1]\), where \(R:={\mathbb {Z}}[\sigma _1,\sigma _2,\tau _1,\ldots , \tau _4,\sigma _2^{1},\tau _4^{1}]\) and hence the group \(\Gamma \) they generate as well. The traces of all elements of the monodromy group are in R.
In particular, in the main case of interest for this paper (the motivic case, see the Introduction) the characteristic polynomials \(q_0,q_1,q_\infty \) will have only roots of unity as roots. In this case, K is a cyclotomic field. We conclude that the monodromy can be conjugated to lie in \({\text {GL}}_4({{\mathcal {O}}}_F)\), where \({{\mathcal {O}}}_F\) is the ring of integers of \(F=K(a_1)\). This is consistent with the rigid local system being motivic.
5 Finite monodromy
We would like to describe all cases of GII with finite monodromy. Since the monodromy is integral (Sect. 4.3), finite monodromy is equivalent to the invariant Hermitian form being definite in every complex embedding of the field of definition. (This is the same argument used in [2].) These cases are those where all solutions to the corresponding differential equation (Sect. 7) are algebraic. Checking definiteness is easily done using the combinatorial criterion of Proposition 2, which involves the relative position of the eigenvalues of the local monodromies.
Apart from the infinitely many imprimitive cases discussed later in Sect. 12.5, the only examples of irreducible cases with finite monodromy that we obtained after an extensive search are those given in Tables 1, 2, 3 and 4.
5.1 Description of the tables
For each choice of eigenvalues, we list the order of the monodromy \(\Gamma \subseteq {\text {GL}}_4({\mathbb {C}})\), an identification of A and the quotient of \(\Gamma /A\) using standard notation (A denotes a maximal abelian normal subgroup of \(\Gamma \)), the order of the center of \(\Gamma \) and whether \(\Gamma \) acts primitively or not.
By a theorem of Jordan, there are finitely many possibilities for the quotient \(\Gamma /A\). The finite groups acting in four dimensions were classified by Blichfeldt (see [11] for a modern description). The group denoted by \(\Gamma _{25920}\) is a simple group.
\(\alpha _1, \alpha _2=1/3,2/3\)
\(\gamma \)  \(\Gamma \)  \(\Gamma /A\)  A  \(Z(\Gamma )\)  \(\text {Impr}\)  

1  1 / 8, 3 / 8, 5 / 8, 7 / 8  48  \(S_4\)  \(C_2\)  2  \(*\) 
2  1 / 5, 2 / 5, 3 / 5, 4 / 5  60  \(A_5\)  1  1  
3  1 / 10, 3 / 10, 7 / 10, 9 / 10  120  \(A_5\)  \(C_2\)  2  
4  1 / 12, 5 / 12, 7 / 12, 11 / 12  144  \(C_2\times A_4\)  \(C_6\)  2  \(*\) 
5  1 / 20, 9 / 20, 13 / 20, 17 / 20  240  \(A_5\)  \(C_4\)  4  
6  2 / 9, 1 / 3, 5 / 9, 8 / 9  324  \(A_4\)  \(C_3^3\)  1  \(*\) 
7  1 / 24, 7 / 24, 17 / 24, 23 / 24  576  \(S_4\times A_4\)  \(C_2\)  2  
8  1 / 28, 9 / 28, 3 / 4, 25 / 28  672  \({\text {PSL}}_2({\mathbb {F}}_7)\)  \(C_4\)  4  
9  1 / 20, 9 / 20, 11 / 20, 19 / 20  720  \(C_2\times A_5\)  \(C_6\)  2  \(*\) 
10  1 / 15, 4 / 15, 11 / 15, 14 / 15  1440  \(A_5\times A_4\)  \(C_2\)  2  
11  1 / 30, 11 / 30, 19 / 30, 29 / 30  1440  \(A_5\times A_4\)  \(C_2\)  2  
12  1 / 40, 9 / 40, 31 / 40, 39 / 40  2880  \(A_5\times S_4\)  \(C_2\)  2 
\(\alpha _1, \alpha _2=1/4,3/4\)
\(\gamma \)  \(\Gamma \)  \(\Gamma /A\)  A  \(Z(\Gamma )\)  \(\text {Impr}\)  

1  1 / 12, 5 / 12, 7 / 12, 11 / 12  192  \(C_2\times S_4\)  \(C_4\)  2  \(*\) 
2  1 / 20, 9 / 20, 13 / 20, 17 / 20  640  \(C_2^4\rtimes D_5\)  \(C_4\)  4  
3  1 / 36, 13 / 36, 25 / 36, 11 / 12  103680  \(\Gamma _{25920}\)  \(C_4\)  4 
\(\alpha _1, \alpha _2=1/5,4/5\)
\(\gamma \)  \(\Gamma \)  \(\Gamma /A\)  A  \(Z(\Gamma )\)  \(\text {Impr}\)  

1  1 / 12, 5 / 12, 7 / 12, 11 / 12  1200  \(C_2\times A_5\)  \(C_{10}\)  2  \(*\) 
2  2 / 15, 7 / 15, 8 / 15, 13 / 15  7200  \(A_5\times A_5\)  \(C_2\)  2  
3  1 / 20, 9 / 20, 11 / 20, 19 / 20  1200  \(C_2\times A_5\)  \(C_{10}\)  2  \(*\) 
4  1 / 30, 11 / 30, 19 / 30, 29 / 30  7200  \(A_5\times A_5\)  \(C_2\)  2 
5.2 Special case
We start by discussing a special, simpler case. Assume that the characteristic polynomials \(q_0,q_1,q_\infty \) of the local monodromies at the respective singularities have real coefficients and that \(q_1=(T1)^2(T+1)^2\). Let \(\gamma _i\in (0,1)\) for \(i=1,\ldots , 4\) be the exponents of the roots of \(q_\infty \) (so that \(c_j=\exp (2\pi i \gamma _j)\)) and similarly let \(\alpha _1 \in (0,1/2)\) be such that the exponents of \(q_0\) are \(0,0,\alpha _1,1\alpha _1\).
General case
\(\beta ,\alpha _1,\alpha _2\)  \( \gamma \)  \( \Gamma \)  \(\Gamma /A\)  A  \(Z(\Gamma )\)  \(\text {Impr}\)  

1  1 / 2, 1 / 2, 1 / 3  1 / 8, 11 / 24, 5 / 8, 23 / 24  4608  \((A_4\times A_4)\rtimes C_2\)  \(C_4^2\)  4  \(*\) 
2  1 / 2, 1 / 2, 1 / 3  5 / 48, 23 / 48, 29 / 48, 47 / 48  41472  \((A_4\times A_4)\rtimes D_4\)  \(C_6^2\)  6  \(*\) 
3  1 / 2, 1 / 2, 1 / 3  11 / 120, 59 / 120, 71 / 120, 119 / 120  1036800  \((A_5\times A_5)\rtimes C_2\)  \(C_{12}^2\)  12  \(*\) 
4  1 / 2, 1 / 2, 1 / 4  7 / 48, 23 / 48, 31 / 48, 47 / 48  6144  \( C_2^4\rtimes D_6\)  \(C_4\cdot C_8\)  8  \(*\) 
5  1 / 2, 1 / 2, 1 / 5  7 / 40, 19 / 40, 27 / 40, 39 / 40  2880000  \((A_5\times A_5)\rtimes C_2\)  \(C_{20}^2\)  20  \(*\) 
6  1 / 2, 1 / 2, 1 / 5  19 / 120, 59 / 120, 79 / 120, 119 / 120  2880000  \((A_5\times A_5)\rtimes C_2\)  \(C_{20}^2\)  20  \(*\) 
7  1 / 2, 1 / 2, 1 / 6  5 / 36, 17 / 36, 29 / 36, 11 / 12  311040  \( \Gamma _{25920}\)  \(C_{12}\)  12  
8  1 / 2, 1 / 2, 1 / 6  11 / 60, 23 / 60, 47 / 60, 59 / 60  311040  \( \Gamma _{25920}\)  \(C_{12}\)  12  
9  1 / 2, 1 / 3, 1 / 4  7 / 24, 5 / 12, 19 / 24, 11 / 12  165888  \((A_4\times A_4)\rtimes D_4\)  \(C_{12}^2\)  12  \(*\) 
10  1 / 2, 1 / 3, 1 / 4  11 / 48, 23 / 48, 35 / 48, 47 / 48  165888  \((A_4\times A_4)\rtimes D_4\)  \(C_{12}^2\)  12  \(*\) 
11  1 / 2, 1 / 3, 1 / 5  4 / 15, 7 / 15, 23 / 30, 29 / 30  6480000  \((A_5\times A_5)\rtimes C_2\)  \(C_{30}^2\)  30  \(*\) 
12  1 / 2, 1 / 3, 1 / 5  17 / 60, 9 / 20, 47 / 60, 19 / 20  6480000  \((A_5\times A_5)\rtimes C_2\)  \(C_{30}^2\)  30  \(*\) 
13  1 / 2, 1 / 3, 1 / 5  19 / 60, 5 / 12, 49 / 60, 11 / 12  6480000  \((A_5\times A_5)\rtimes C_2\)  \(C_{30}^2\)  30  \(*\) 
14  1 / 2, 1 / 3, 1 / 5  29 / 120, 59 / 120, 89 / 120, 119 / 120  6480000  \((A_5\times A_5)\rtimes C_2\)  \(C_{30}^2\)  30  \(*\) 
15  1 / 2, 1 / 5, 2 / 5  4 / 15, 13 / 30, 23 / 30, 14 / 15  6000  \(S_5\)  \(C_2\cdot C_{5}^2\)  10  \(*\) 
16  1 / 2, 1 / 5, 2 / 5  9 / 40, 19 / 40, 29 / 40, 39 / 40  6000  \(S_5\)  \(C_2\cdot C_{5}^2\)  10  \(*\) 
17  1 / 3, 1 / 2, 5 / 6  1 / 18, 7 / 18, 13 / 18, 5 / 6  155520  \(\Gamma _{25920}\)  \(C_6\)  6  
18  1 / 3, 1 / 2, 1 / 6  5 / 18, 11 / 18, 5 / 6, 17 / 18  155520  \(\Gamma _{25920}\)  \(C_6\)  6  
19  1 / 3, 1 / 2, 1 / 6  11 / 30, 17 / 30, 23 / 30, 29 / 30  155520  \(\Gamma _{25920}\)  \(C_6\)  6  
20  1 / 3, 1 / 3, 2 / 3  1 / 12, 11 / 24, 5 / 6, 23 / 24  69120  \(C_2^4.A_6\)  \(C_{12}\)  12  
21  1 / 3, 1 / 3, 2 / 3  2 / 15, 8 / 15, 11 / 15, 14 / 15  2160  \(A_6\)  \(C_6\)  6  
22  1 / 3, 1 / 3, 2 / 3  5 / 24, 11 / 24, 17 / 24, 23 / 24  2160  \(A_6\)  \(C_6\)  6  
23  1 / 3, 1 / 3, 2 / 3  5 / 42, 17 / 42, 5 / 6, 41 / 42  15120  \(A_7\)  \(C_6\)  6  
24  1 / 3, 1 / 3, 2 / 3  11 / 60, 23 / 60, 47 / 60, 59 / 60  69120  \( C_2^4.A_6\)  \(C_{12}\)  12 
Proposition 3
With the above assumptions and notations the invariant Hermitian form H is definite if and only if \((n_1,n_2)=(2,4)\).
In the special case of this section, the finite monodromy cases found are listed in Tables 1, 2, and 3.
5.3 General case
6 Coxeter groups
To find explicit realizations of finite monodromy groups of GII type, we may start with a finite group in \({\text {GL}}_4({\mathbb {C}})\) and attempt to build a GII rigid local system by producing three appropriate elements \(T_0,T_1,T_\infty \). For example, we can take a finite complex reflection group W in rank 4, hence one of the Weyl groups \(A_4,B_4,F_4\) or the noncrystallographic case \(H_4\). Since \(T_\infty \) should have distinct eigenvalues different from 1, we could start by taking \(T_\infty \) to be a Coxeter element. Similarly, we can take \(T_1\) to be the product of two commuting reflections in W. We may assume that these reflections are simple and hence correspond to two nonadjacent dots in the corresponding Dynkin diagram.
7 Differential equation
Signature (2, 2)
\(\alpha _1,\alpha _2\)  \(\beta _1,\beta _2\)  \(\gamma _1,\gamma _2,\gamma _3,\gamma _4\)  D  \(\mu \)  

1  1 / 3, 2 / 3  0, 1 / 2  1 / 4, 1 / 3, 2 / 3, 3 / 4  \(\)3  \(\)2 
2  1 / 3, 2 / 3  0, 1 / 2  1 / 6, 1 / 4, 3 / 4, 5 / 6  \(\)3  \(\)2 
3  1 / 3, 2 / 3  1 / 3, 2 / 3  1 / 6, 1 / 4, 3 / 4, 5 / 6  \(\)3  \(\)2 
4  1 / 3, 2 / 3  1 / 3, 2 / 3  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)3  \(\)1 
5  1 / 3, 2 / 3  1 / 6, 5 / 6  1 / 4, 1 / 3, 2 / 3, 3 / 4  \(\)3  \(\)2 
6  1 / 3, 2 / 3  1 / 6, 5 / 6  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)3  \(\)1 
7  1 / 4, 3 / 4  0, 1 / 2  1 / 4, 1 / 3, 2 / 3, 3 / 4  \(\)4  \(\)3 
8  1 / 4, 3 / 4  0, 1 / 2  1 / 6, 1 / 3, 2 / 3, 5 / 6  \(\)4  \(\)1 
9  1 / 4, 3 / 4  0, 1 / 2  1 / 6, 1 / 4, 3 / 4, 5 / 6  \(\)4  \(\)3 
10  1 / 4, 3 / 4  0, 1 / 2  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)4  \(\)1 
11  1 / 4, 3 / 4  0, 1 / 2  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)4  \(\)1 
12  1 / 4, 3 / 4  1 / 3, 2 / 3  1 / 6, 1 / 4, 3 / 4, 5 / 6  \(\)4  \(\)3 
13  1 / 4, 3 / 4  1 / 3, 2 / 3  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)4  \(\)1 
14  1 / 4, 3 / 4  1 / 3, 2 / 3  1 / 12, 5 / 12, 7 / 12, 11 / 12  \(\)4  1 
15  1 / 4, 3 / 4  1 / 4, 3 / 4  1 / 12, 5 / 12, 7 / 12, 11 / 12  \(\)4  1 
16  1 / 4, 3 / 4  1 / 6, 5 / 6  1 / 4, 1 / 3, 2 / 3, 3 / 4  \(\)4  \(\)3 
17  1 / 4, 3 / 4  1 / 6, 5 / 6  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)4  \(\)1 
18  1 / 4, 3 / 4  1 / 6, 5 / 6  1 / 12, 5 / 12, 7 / 12, 11 / 12  \(\)4  1 
19  1 / 6, 5 / 6  0, 1 / 2  1 / 4, 1 / 3, 2 / 3, 3 / 4  \(\)3  \(\)2 
20  1 / 6, 5 / 6  0, 1 / 2  1 / 6, 1 / 3, 2 / 3, 5 / 6  \(\)3  \(\)2 
21  1 / 6, 5 / 6  0, 1 / 2  1 / 6, 1 / 4, 3 / 4, 5 / 6  \(\)3  \(\)2 
22  1 / 6, 5 / 6  0, 1 / 2  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)3  \(\)1 
23  1 / 6, 5 / 6  0, 1 / 2  1 / 8, 3 / 8, 5 / 8, 7 / 8  \(\)3  \(\)1 
24  1 / 6, 5 / 6  0, 1 / 2  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)3  \(\)1 
25  1 / 6, 5 / 6  1 / 3, 2 / 3  1 / 6, 1 / 4, 3 / 4, 5 / 6  \(\)3  \(\)2 
26  1 / 6, 5 / 6  1 / 3, 2 / 3  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)3  1 
27  1 / 6, 5 / 6  1 / 3, 2 / 3  1 / 8, 3 / 8, 5 / 8, 7 / 8  \(\)3  1 
28  1 / 6, 5 / 6  1 / 3, 2 / 3  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)3  1 
29  1 / 6, 5 / 6  1 / 3, 2 / 3  1 / 12, 5 / 12, 7 / 12, 11 / 12  \(\)3  2 
30  1 / 6, 5 / 6  1 / 4, 3 / 4  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)3  1 
31  1 / 6, 5 / 6  1 / 4, 3 / 4  1 / 8, 3 / 8, 5 / 8, 7 / 8  \(\)3  1 
32  1 / 6, 5 / 6  1 / 4, 3 / 4  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)3  1 
33  1 / 6, 5 / 6  1 / 4, 3 / 4  1 / 12, 5 / 12, 7 / 12, 11 / 12  \(\)3  2 
34  1 / 6, 5 / 6  1 / 6, 5 / 6  1 / 4, 1 / 3, 2 / 3, 3 / 4  \(\)3  \(\)2 
35  1 / 6, 5 / 6  1 / 6, 5 / 6  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)3  1 
36  1 / 6, 5 / 6  1 / 6, 5 / 6  1 / 8, 3 / 8, 5 / 8, 7 / 8  \(\)3  1 
37  1 / 6, 5 / 6  1 / 6, 5 / 6  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)3  1 
38  1 / 6, 5 / 6  1 / 6, 5 / 6  1 / 12, 5 / 12, 7 / 12, 11 / 12  \(\)3  2 
8 Field of moduli \({\mathbb {Q}}\)
Signature (4, 0)
\(\alpha _1,\alpha _2\)  \(\beta _1,\beta _2\)  \(\gamma _1,\gamma _2,\gamma _3,\gamma _4\)  D  \(\mu \)  

1  1 / 3, 2 / 3  0, 1 / 2  1 / 5, 2 / 5, 3 / 5, 4 / 5  \( 3\)  1 
2  1 / 3, 2 / 3  0, 1 / 2  1 / 8, 3 / 8, 5 / 8, 7 / 8  \(\)3  1 
3  1 / 3, 2 / 3  0, 1 / 2  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)3  1 
4  1 / 3, 2 / 3  0, 1 / 2  1 / 12, 5 / 12, 7 / 12, 11 / 12  \(\)3  2 
5  1 / 3, 2 / 3  1 / 3, 2 / 3  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)3  \(\)1 
6  1 / 3, 2 / 3  1 / 3, 2 / 3  1 / 8, 3 / 8, 5 / 8, 7 / 8  \(\)3  \(\)1 
7  1 / 3, 2 / 3  1 / 4, 3 / 4  1 / 6, 1 / 3, 2 / 3, 5 / 6  \(\)3  \(\)2 
8  1 / 3, 2 / 3  1 / 4, 3 / 4  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)3  \(\)1 
9  1 / 3, 2 / 3  1 / 4, 3 / 4  1 / 8, 3 / 8, 5 / 8, 7 / 8  \(\)3  \(\)1 
10  1 / 3, 2 / 3  1 / 4, 3 / 4  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)3  \(\)1 
11  1 / 3, 2 / 3  1 / 6, 5 / 6  1 / 8, 3 / 8, 5 / 8, 7 / 8  \(\)3  \(\)1 
12  1 / 3, 2 / 3  1 / 6, 5 / 6  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)3  \(\)1 
13  1 / 4, 3 / 4  0, 1 / 2  1 / 12, 5 / 12, 7 / 12, 11 / 12  \(\)4  1 
14  1 / 4, 3 / 4  1 / 3, 2 / 3  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)4  \(\)1 
15  1 / 4, 3 / 4  1 / 4, 3 / 4  1 / 6, 1 / 3, 2 / 3, 5 / 6  \(\)4  \(\)1 
16  1 / 4, 3 / 4  1 / 4, 3 / 4  1 / 5, 2 / 5, 3 / 5, 4 / 5  \(\)4  \(\)1 
17  1 / 4, 3 / 4  1 / 4, 3 / 4  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)4  \(\)1 
18  1 / 4, 3 / 4  1 / 6, 5 / 6  1 / 10, 3 / 10, 7 / 10, 9 / 10  \(\)4  \(\)1 
Quaternion algebras
\(D\backslash \mu \)  \( 3 \)  \( 2\)  \(1\)  1  2 

\(3\)  \( [3,\infty ]\)  \([2,\infty ]\)  \([3,\infty ]\)  [ ]  [2, 3] 
\(4\)  \( [3,\infty ]\)  \([2,\infty ]\)  \([2,\infty ]\)  [ ] 
9 Finite monodromy \(K={\mathbb {Q}}\)
As shown in Table 6, there are only four cases of finite monodromy with field of moduli \({\mathbb {Q}}\) that can be realized over \({\mathbb {R}}\) (rows #1, #2, #4, and #14; note that #3 is a twist of #1). Three cases are actually definable over \({\mathbb {Q}}\); we list these first. We give the fourth case in Sect. 9.4; it has the quaternion algebra ramified at [2, 3] as an obstruction and is hence not definable over \({\mathbb {Q}}\).
We will construct these monodromy groups as subgroups in Coxeter groups as in Sect. 6; we circle in the corresponding Dynkin diagram the two chosen simple reflections.
9.1 \((1/3,2/3),(0,1/2),(1/5,2/5,3/5,4/5)\)
9.2 \((1/4,3/4),(0,1/2),(1/8,3/8,5/8,7/8)\)
9.3 \((1/4,3/4),(0,1/2),(1/12,5/12,7/12,11/12)\)
9.4 \((1/3,2/3),(0,1/2),(1/12,5/12,7/12,11/12)\)
As already mentioned in this case, the quaternion algebra is ramified at \([2,3]\). As it happens, since 2 and 3 are both inert in \(F={\mathbb {Q}}(\sqrt{5})\), the obstruction cocycle \(\xi \) becomes trivial in \(F\). Therefore, by Proposition 3, we should be able to realize this case over \(F\).
This is indeed the case and we can realize it again using Coxeter groups, namely as a subgroup of the noncrystallographic \(W(H_4)\) of order 14400. Note, however, that in this case \(T_\infty \) is not a Coxeter element; we still take \(T_1\) as a product of two commuting simple reflections.
10 Hurwitz example
Some examples of GII systems arise from modular functions. We give an example due to Hurwitz [15]. For more details on the associated Klein curve, see [7]; for general facts about modular forms, see [28].
11 A family of genus two curves
In this section, we analyze in depth the GII system in row \(\# 35\) of Table 5, which has infinite monodromy group. We show explicitly that it is motivic by matching it to a Picard–Fuchs equation of an associated family of genus two curves.
We will show that \({{\mathcal {G}}}\) arises from \(H^1\) of a family of genus two curves (so it is motivic). To find these, we use an argument we learned from D. Roberts. We will see that the group \(\Gamma \) equals the monodromy of a finite monodromy GII modulo 2 (denoted \({{\mathcal {G}}}_1\) below) and use it to produce a family of polynomials of degree 6 which give rise to the desired curves.
Note that the parameters for \({{\mathcal {G}}}\) and \({{\mathcal {G}}}_1\) are equal up to fractions with denominator 2. This means that their respective local monodromies are the same modulo 2. It is clear, for example, that the monodromy of \({{\mathcal {G}}}_1\) is isomorphic to \(A_6\cong {\text {PSL}}_2({\mathbb {F}}_9)\) modulo 2 as the center acts by \(\pm 1\).
We check that this indeed the case by computing the linear differential equation satisfied by periods of \(C_t\). Starting with \(\omega :=dx/y\) we apply \( D:=d/dt.\) reducing at each stage to a representative differential form of the type p(x) / ydx with p of degree at most five modulo exact differentials. We then look for a linear relation among \(\omega ,D\omega \ldots , D^4\omega \).
12 Infinite families
Considering the stringent conditions required for the invariant Hermitian form H to be definite, it can seem unlikely that there would be infinitely many examples where H and all its Galois conjugates are definite. However, just as for hypergeometrics [2, Theorem 5.8], this is indeed the case. Moreover, again like hypergeometrics, they come in families all of which have the (finite) monodromy group \(\Gamma \subseteq {\text {SL}}_4({\mathbb {C}})\) acting imprimitively (see Table 8). This is not surprising in light of Jordan’s theorem (see the discussion at the end of Sect. 5.1).
Infinite families
\(\beta ,\alpha _1,\alpha _2\)  \( \gamma \)  \(\Gamma /A\)  A  \(Z(\Gamma )\)  Impr  

1  1 / 2, 1 / 2, r  \(r/4,1/4r/4,1/2r/4,3/4r/4 \)  \(D_4\)  \(\Delta _{1,n}\)  n  \(*\) 
2  1 / 2, 1 / 3, 2 / 3  \(r,r/3,1/3r/3,2/3r/3 \)  \(A_4\)  \(\Delta _{2,n}\)  \(\hbox {gcd}(n,4)\)  \(*\) 
We will show that in fact there are such examples for all of the cases considered by Simpson [24] with \(g=0,k=3\) punctures and one partition equal to \(1^n\) for some n. For all of these systems, there are infinite families of examples all lying in a single geodesic in the positive components \({\mathbb {T}}_+^{{\text {irr}}}\).
12.1 Rational powers
We start by showing that the rational powers of algebraic functions satisfy differential equations of certain fixed order.
Proposition 4
Let f(t) be an algebraic function of degree m. Then for all \(r\in {\mathbb {Q}}\) the function \(f^r\) satisfies \(\mathcal {L}_rf^r = 0\), where \(\mathcal {L}_r\) is a differential operator of order m, whose coefficients depend polynomially on r.
Proof
For generic r (more precisely, whenever \(W(y_1^r,\dots ,y_m^r)\ne 0\)) the functions \(y_1^r,\dots ,y_m^r\) form the full space of solutions of \(\mathcal {L}_r\), and thus, the singularities of \(\mathcal {L}_r\) are contained in the set of singularities of \(y_1^r,\dots ,y_m^r\) together with the set of points t where one of \(y_i\) becomes 0. In terms of the defining equation \(P(t,y)=p_m(t)y^m+\dots +p_0(t)\), these are exactly the values of t where \(p_0(t)p_m(t)\) or the discriminant of P vanishes. \(\square \)
12.2 Simpson even and odd families
At a zero or pole of \(f_N(t)\) we have \(t=0,1\) or \(t=\infty \). Hence by Proposition 4, \(f_N(t)^r\) for \(r\in {\mathbb {Q}}\) satisfies a linear differential equation of order N with singularities only at \(t=0,1,\infty \).
In general, the exponents at \(t=0,\infty \) of the differential equation satisfied by an algebraic function f of this kind can be readoff from its Newton polygon \(\Delta \). It can be proved that these are as follows.
Assume that the Newton polygon of f has no vertical segments. Then there exist unique leftmost and rightmost vertices of \(\Delta \), say p, q, respectively. Let l be the line joining p and q. We can distinguish the top and bottom sides of \(\Delta \) as those above and below l, respectively.
The exponents at \(t=0\) for \(f^r\) are \([\kappa _1],[\kappa _2],\ldots \), where \(\kappa _1,\kappa _2,\ldots \) runs over the slopes of the bottom sides. The exponents at \(t=\infty \) are similarly determined by the slopes of the top sides. The exponents at \(t=1\) are independent of r and can be computed directly from the Newton polygon of \(p(u,t+1)\).
12.3 Simpson extra case of rank 6
12.4 Hypergeometric
Theorem 5.8 in [2] describes a geodesic in the case of hypergeometric rigid local systems. This can be made explicit in terms of fractional powers of a fixed algebraic function like all the previous examples. We have already encountered one case (see the example at the end of Sect. 12.1). We illustrate this further with an instance of rank 5.
12.5 Goursat II
Notes
Declarations
Acknowledgements
Open access funding provided by Max Planck Society. This work was started at the Abdus Salam Centre for Theoretical Physics and completed during the special trimester Periods in Number Theory, Algebraic Geometry and Physics at the Hausdorff Institute of Mathematics in Bonn, Germany. We would like to thank these institutions as well as the Max Planck Institute for Mathematics in Bonn for their hospitality and support. We thank the anonymous referee for helpful comments. The second author would like to thank N. Katz and D. Roberts for useful exchanges regarding the subject of this work.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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