Goursat rigid local systems of rank four

We study the general properties of certain rank four rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form 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 the rationals. We also exhibit one example with infinite monodromy as arising from a family of genus two curves.


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 1800's. In his 1897 thesis written under the supervision of P. Painlevé [4], [15], [22], A. Boulanger mentions a paper by J. Liouville of 1833 [14] 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.
Schwarz [19] famously described all cases of algebraic solutions to the hypergeometric equation satisfied by Gauss's series 2 F 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.
From a broader point of view, we may say that differential equations with all solutions algebraic are a special case of motivic local systems. Simpson conjectures in [21, p. 9] that all rigid local systems satisfying some natural conditions are motivic. Without attempting a rigorous definition of what this means, we will just say that such systems should be geometric in nature. This is known for rigid local systems on P 1 by the work of Katz [12], who gave a general algorithm (using middle convolution) for their construction. See also [7] for systems over a higher dimensional base and [24] for more on differential equations and arithmetic.
In this note we consider the case of Goursat's case II of rank 4 rigid local systems (denoted henceforth by G-II). These correspond to order 4 linear differential equations with three regular singular points, say t = 0, 1, ∞, with semisimple, finite order local monodromies with eigenvalues of multiplicities 21 2 , 2 2 , 1 4 respectively.
We study the general properties of G-II systems; for example, we analyze when they are irreducible and describe the invariant Hermitian form H when it exists. As in [2] H is a key tool to understand when the monodromy 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 that the monodromy group can be defined in an integral way in terms of the eigenvalues of the local monodromies (the defining data). This would follow if the rigid system was motivic (see [21, p. 9]). We find (see §3) that there is a non-trivial 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 G-II systems the obstruction is given by a quaternion algebra over the field of moduli.
By a computer search we find what we expect are all irreducible G-II equations whose solutions are algebraic functions and give several explicit examples defined over Q. We also exhibit one example with infinite monodromy as arising from a family of genus two curves.
We present in this paper our results with few detailed proofs, which will appear in a subsequent work. We used MAGMA [3] and PARI-GP [16] for most of the calculations.

Acknowledgements
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.
The second author would like to thank N. Katz and D. Roberts for useful exchanges regarding the subject of this work.

Rigid local systems
Following the setup and notation of [10] we consider the character variety M µ where µ is an ordered k-tuple of partitions of a positive integer n. This variety parametrizes representations of π 1 (Σ \ S, * ) to GL n (C) mapping a small oriented loop around s ∈ S to a semisimple conjugacy class C s whose generic eigenvalues have multiplicities µ s = (µ s 1 , µ s 2 , . . .), a corresponding partition in µ. Here Σ is a Riemann surface of genus g and S is a finite set of k points. The eigenvalues are assumed generic in the sense of [10]. If non-empty the variety M µ is equidimensional of dimension d µ := (2g − 2 + k)n 2 − s∈S i≥1 (µ s i ) 2 + 2.
In this paper we will only consider the case where g = 0 and in detail when k = 3, n = 4 and, taking S = {0, 1, ∞}, the partitions are µ 0 = 21 2 , µ 1 = 2 2 , µ ∞ = 1 4 . To be concrete, if g = 0, given conjugacy classes C 1 , . . . , C k ⊆ GL n (C) and labeling the punctures with 1, . . . , k, we are looking for solutions to (1) T 1 · · · T k = I n , T s ∈ C s , s = 1, . . . , k, where I n is the identity matrix, up to simultaneous conjugation. Given such a representation π 1 (Σ \ S, * ) we call the image in Γ := T 1 , . . . , T k ≤ GL n (C) the geometric monodromy group. It is well defined up to conjugation. Goursat in his remarkable 1886 paper [8] 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 k-tuples of matrices T s ∈ C s and T s ∈ C s for s ∈ S satisfying (1) there exists a single U ∈ GL n (C) such that T s = U T s U −1 for all s ∈ S. The corresponding local systems (determined by the local solutions to the linear differential equation) are known as rigid local systems [12].
To have a rigid local system is to say that M µ consists of a single point. Therefore it is necessary that the expected dimension d µ be zero. This is precisely Goursat's condition [8, (5) p.113] (he only considers the case of g = 0) as well as Katz's [12], which follows from cohomological considerations.
We assume from now on that g = 0 and then to avoid trivial cases that k ≥ 3. Indeed, for g = 0, k = 1 the group π 1 (Σ \ S, * ) is trivial and for g = 0, k = 2 it is isomorphic to Z. Note, as Goursat points out, that adding an extra puncture to S with associated partition (n) does not change the value of d µ . Such points correspond to apparent singularities in the differential equation and may hence be safely ignored. We will assume then that the partitions µ s have at least two parts.
Goursat shows that with the given assumptions k ≤ n + 1 [8, top p.114] and hence there are only finitely many solutions of d µ = 0 for fixed n. He lists [8, p.115] the cases of d µ = 0 for n = 3 and n = 4 (see below).
It turns out, however, that the condition d µ = 0 is not sufficient as the variety M µ might be empty. Crawley-Boevey [5] proved that a necessary and sufficient condition for M µ to be a point, in the case of generic eigenvalues we are considering, is that µ 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 µ represents a real root. This algorithm ultimately corresponds to Katz's middle convolution.
It is more convenient to present the multiplicity data µ in the form of a star graph with one central node and k legs. We illustrate this in our basic case G-II (Goursat's label II for n = 4).
The partitions can be read by starting at the central node and moving away along a leg. The succesive differences of the respective node values are the parts of the corresponding partition. Nodes with a zero value are not included; hence each leg has a finite length equal to that of the corresponding partition. The algorithm proceeds starting from a configuration as above corresponding to an ordered k-tuple µ of partitions of n satisfying d µ = 0 using the following moves.
• A: Replace the value n at the central node by where n i are the values at the nodes closest to the central node. • 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 non-decreasing 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). The goal is to use a sequence of these moves to reach the terminal configuration of just a central node with value 1. Under the assumptions d µ = 0, g = 0 applying A strictly decreases the value at the central node and hence the algorithm always terminates. Indeed, for any partition µ = µ 1 ≥ µ 2 ≥ · · · of n we have nµ 1 and since also g = 0 and n > 0 that s µ s 1 > (k − 2)n which proves the claim. For our running example µ = (21 2 , 2 2 , 1 4 ) the algorithm works as follows.
Apply A:  2   1  2  3  3  2  1   Apply B:   2   1  2  3  2  1 Apply C: We have arrived at the case µ = (1 3 , 21, 1 3 ) that corresponds to the hypergeometric equation of order 3. It is easy to see that a next stage takes us to µ = (1 2 , 1 2 , 1 2 ) corresponding to the hypergeometric equation of rank 2 and finally to the desired terminal case. This confirms that indeed G-II corresponds to a rigid local system. The algorithm fails if at any stage we cannot perform C; i.e. applying A yields a graph with a central value strictly smaller than one of its neighbors. As Goursat points out this happens for his case IV.
Apply A: Since 2 < 3 we cannot apply C on the leg going off to the right (one of the parts would have to be 2 − 3 = −1).
Here are the diagrams of all rank n = 4 rigid local systems of the type in question and their corresponding label in Goursat's paper (all but the case IV just discussed actually correspond to a rigid local system).

Field of definition and field of moduli
Given a rigid local system with conjugacy classes C s for s ∈ S as in §2 let q s (T ) be the characteristic polynomial of any element of C s . Let K be the field obtained by adjoining to 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 ∈ GL n (F ) of the required kind, i.e., T s ∈ 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 Q containing all of their entries. We necessarily have K ⊆ 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 §4.3).
Let T s ∈ C s be a k-tuple of matrices in GL n (Q) satisfying (1). It is clear that for σ ∈ Gal(Q/K) the k-tuple T σ s is another solution to (1). Hence by rigidity there exists X σ ∈ GL n (Q) such that Again by rigidity we find that there exists a σ,τ ∈Q such that X σ X σ τ = a σ,τ X στ . The map (σ, τ ) → a σ,τ is a 2-cocycle giving a well defined element ξ ∈ H 2 (Gal(K/K), K × ).
The following is a standard result.

Proposition 1.
There exists a solution to (1) over K if and only if ξ 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 Q containing the traces of all T ∈ Γ.

Explicit solution for the Goursat case II
In [8] Goursat writes down an explicit solution to (1) for S = {0, 1, ∞} in the case when T 0 , T 1 , and T ∞ are diagonalizable with spectra 1 2 a 1 a 2 , 1 2 b 2 and c 1 c 2 c 3 c 4 respectively (assuming that eigenvalues with different labels are distinct and that a 1 a 2 b 2 c 1 c 2 c 3 c 4 = 1).
Since the triple (T 0 , T 1 , T ∞ ) is irreducible, the 1-eigenspaces for T 0 and T 1 must have a zero intersection. Goursat then shows that in a suitable basis the matrices T 0 and T 1 are given by A direct computation shows that for given a 1 , a 2 , and b, the coefficients of q ∞ depend linearly on A, D, and AD − BC. Conversely, the numbers A, D, and AD − BC can be found from q ∞ by In particular, these identities imply that A, D, and BC are uniquely determined from the spectra. On the other hand, conjugation by the diagonal matrix D = diag(λ, 1, λ, 1) preserves the shapes of T 0 and T 1 and maps (B, C) to (λ −1 B, λC), hence only the product BC is uniquely determined.
4.1. Criterion for irreducibility. We now find a criterion for when the constructed representation is irreducible. The eigenmatrices for T 0 and T 1 are To get the description in terms of eigenvalues we use the following factorizations: Note that in terms of q ∞ this simply becomes q ∞ (1) = 0, q ∞ (b −1 ) = 0, and is the polynomial whose roots are products of all pairs of roots of q ∞ . This description agrees with the conditions given in [17, p. 10].

4.2.
Invariant Hermitian form. Let T 1 , . . . , T k correspond to an irreducible local system. Assume that there exists a non-zero Hermitian form H invariant under Γ.
Since ker(H) is invariant under all T s by irreducibility we get that any such H must be non-degenerate. This implies, in particular, that (T s * ) −1 and T s are conjugate. Therefore the sets of eigenvalues of T s are invariant under the map z →z −1 . This is certainly the case if the eigenvalues are in the unit circle.
On the other hand, if the eigenvalues of T s are invariant under the map z →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 geometric monodromy group. The set T 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 α 1 < α 2 and γ 1 < γ 2 < γ 3 < γ 4 . Then there is a unique connected component T irr + where H is positive definite. We can compute the invariant form explicitly starting from (3). The equations (4) imply in this case that A, D, and BC are real. After making a suitable conjugation for (B, C), we may assume that A, B, C, D are real numbers. The invariant Hermitian matrix is then Using (7) we can easily describe T irr + in terms of the parameters (A, D, t) where t = BC. If we look at the connected components of the set , and compute the signature in each case, we find that H is positive definite if and only if To derive a criterion in terms of eigenvalues requires more work, but can be done similarly. The final criterion is then the following. Let I 1 be the open arc in S 1 with end points 0, and b −1 (any of the two possibilities), and let I 2 be the arc with end points where among the two arcs we pick the one that contains the point b −1 .

Proposition 2. The invariant
Hermitian form H is definite if and only if for some labeling c 1 , . . . , c 4 of the eigenvalues of T ∞ we have

4.3.
Integrality. We would like to give an integral form of our local monodromies. The first observation is that we may choose T ∞ as the companion form of q ∞ since it has no repeated roots. After some experimentation we found we can choose T 1 as follows.
With these, using that obtained by taking determinants in T 0 T 1 T ∞ = I 4 , we get The trace field is generically given by K = Q(σ 1 , σ 2 , τ 1 , . . . , τ 4 ) and we see that we can always take as field of definition the quadratic extension F := K(a 1 ). Note that we also have tr(T 0 ) = 2 + a 1 + a 2 ∈ K. Hence a 1 and a 2 are conjugate over K.
In fact, the local monodromies are definable over the ring and hence the group Γ they generate as well. The traces of all elements of the geometric monodromy group are in R.
In particular, in the main case of interest the characteristic polynomials q 0 , q 1 , q ∞ will have only roots of unity as roots. In this case K is a cyclotomic field. We conclude that the geometric monodromy can be conjugated to lie in GL 4 . This is consistent with the rigid local system being motivic.
For example, where ζ 12 is a primitive 12-root of unity. This corresponds to row #3 in Table 2. Then our choice gives and These matrices generate a group Γ of order 103680, which is a non-split central extension by C 4 of the simple group Γ 25920 . We see here a phenomenon that occurs frequently in our examples. The quotient Γ/Z(Γ) has no irreducible representation of degree 4 (the smallest non-trivial irreducible representation is of order 5), whereas a central extension, namely Γ, does. It follows from the above discussion that for G-II cases the cocycle obstruction of §3 is generically of order dividing 2 = [F : K]. We can easily compute the corresponding matrix X σ for σ the generator of Gal(F/K) as in §3. The problem is linear: we solve The cocycle can be represented by a quaternion algebra. Explicitly, this is the quaternion algebra D,µ K , where D = disc(F ) and µ is as above.

Finite monodromy
We would like to describe all cases of G-II with finite geometric monodromy. Since the geometric monodromy is integral §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 §7 are algebraic.
Apart from the infinitely many imprimitive cases discussed later in §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 below. 5.1. Description of the tables. For each choice of eigenvalues we list the order of the geometric monodromy Γ ⊆ GL 4 (C), an identification of A and the quotient of Γ/A using standard notation (A denotes a maximal abelian normal subgroup of Γ), the order of the center of Γ and whether Γ acts primitively or not.
By a theorem of Jordan there are finitely many possibilities for the quotient Γ/A. The finite groups acting in four dimensions were classified by Blichfeldt (see [9] for a modern description). The group denoted by Γ 25920 is a simple group.
We should note that we can always twist the local monodromies by multiplying by scalars matrices so that the resulting triple is in SL 4 (C). If the group acts primitively, the normal subgroup A consists of scalars. It follows that there are finitely many possible primitive Γ up to twisting; we will see in §12 that this is not the case for imprimitive groups.
A special case of Proposition 2 reduces in this case to the following. Let δ 1 , . . . , δ 6 be representatives in (0, 1) (with multiplicities) of the exponents γ i + γ j for i < j. Define n 1 as the number of γ i in the interval (0, 1/2) and n 2 the number of δ i (counting with multiplicities) in the interval (1/2 − α 1 , 1/2 + α 1 ).  In the special case of this section the finite monodromy cases found are listed in Tables 1,2,3 below.
The finite monodromy cases found are listed in Table 4.  Table 4. General case

Coxeter groups
We may start with a finite group in GL 4 (C) and attempt to build a G-II rigid local system by producing three appropriate elements T 0 , T 1 , T ∞ . 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 non-crystallographic case H 4 . Since T ∞ should have distinct eigenvalues different from 1, we could start by taking T ∞ 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 non-adjacent dots in the corresponding Dynkin diagram.
We illustrate the procedure in one example in the case H 4 and give in section §9 examples defined over Q. The Dynkin diagram is where we have circled the two chosen simple reflections. We take T ∞ := s 1 s 2 s 3 s 4 , T 1 := s 1 s 3 and T 0 so that T 0 T 1 T ∞ = 1.

Field of moduli Q
It is easy to list all cases of irreducible G-II rigid local systems with field of moduli Q as there are only finitely many cyclotomic polynomials of fixed degree and with coefficients in Q. The resulting cases are listed in Table 5 and Table 6 according to the signature of the respective Hermitian form.

Finite monodromy K = Q
As we see in the above table there are only four cases. Three cases are actually definable over Q; we list these first. We give the fourth case in §9.4; it has the quaternion algebra ramified at [2,3] as an obstruction and is hence not definable over Q.
A basis of solutions to (11) is then xyz, x 2 y, y 2 z, z 2 x. Note that u 7 = xyz. This is basically given in the original paper of Hurwitz [11]. For more details on the associated Klein curve see [6].

A family of genus two curves
Consider the G-II rigid local system G with parameters (α 1 , α 2 ) = (1/6, 5/6), (β 1 , β 2 ) = (1/6, 5/6), (γ 1 , γ 2 , γ 3 , γ 4 ) = (1/5, 2/5, 3/5, 4/5) and trace field Q. We see from row #35 of Table 5 that the obstruction vanishes and hence it is definable over Q. We find the following concrete realization Computing the invariant Hermitian form we find that these matrices are symplectic. Let Γ := T 0 , T 1 , T ∞ ⊆ Sp 4 (Z) be the geometric monodromy group. We will show that 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 Γ equals the geometric monodromy of a finite monodromy G-II modulo 2 (denoted G 1 below) and use it to produce a family of polynomials of degree 6 which give rise to the desired curves.
Bender [1] has given the following generators for the symplectic group Sp 4 (Z).
Its trace field is Q and indeed we find it in row #5 of Table 6 among those with finite monodromy. Since µ = −1 the system is not definable over Q. Using a realization over Q( √ −3) with MAGMA we find that the geometric monodromy group is isomorphic to SL 2 (F 9 ), a central extension of A 6 by C 2 . This group has two irreducible representations of degree four with Schur index two.
The three even permutations we computed above generate A 6 and correspond to a Belyi map with cycle type 31 3 , 3 2 , 51. D. Roberts showed us how this map is given by the following polynomial Indeed we have We consider the family of genus two curves defined by the hyperelliptic equation Its Igusa invariants are J 2 = 2 2 · 3 · 5 2 · (4t + 1) By construction the Galois representation on the two torsion of the Jacobian of C t for a generic t ∈ Q is congruent modulo two to that of the Artin representation associated to the Belyi map. We therefore expect that the motive H 1 (C t , Q) corresponds to G.
We check that this indeed the case by computing the linear differential equation satisfied by periods of C t . Starting with ω := 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 ω, Dω . . . , D 4 ω.

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) geometric monodromy group Γ ⊆ SL 4 (C) acting imprimitively (see Table 8). This is not surprising in light of Jordan's theorem (see the discussion at the end of §5.1).
In this table r = m/n is an arbitrary rational number in lowest terms and where v 2 is the valuation at 2.
Proposition 4. Let f (t) be an algebraic function of degree m. Then for all r ∈ Q the function f r satisfies L r f r = 0, where L r is a differential operator of order m, whose coefficients depend polynomially on r.
Define polynomial differential operators D n by Then D 0 (f ) = 1, D 1 (f ) = f , D 2 (f ) = f +f 2 , and in general they are defined by the recursion D n+1 (f ) = (D n (f )) + D n (f )f . In terms of these operators we can write the Wronskian in terms of logarithmic derivatives as W (f 1 , . . . , Expanding the determinants in the definition of L r shows that A k can be expressed as rational functions in r whose coefficients are symmetric expressions in y 1 , . . . , y m and their derivatives and thus are rational functions of t. For generic r (more precisely, whenever W (y r 1 , . . . , y r m ) = 0) the functions y r 1 , . . . , y r m form the full space of solutions of L r , and thus the singularities of L r are contained in the set of singularities of y r 1 , . . . , y r m 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 + · · · + p 0 (t), these are exactly the values of t where p 0 (t)p m (t) or the discriminant of P vanishes.
For instance, consider u 2 (τ ) := (η(2τ )/η(τ )) 24 . It is known that u 2 is a Hauptmodul for Γ 0 (2) and satisfies the algebraic equation where j is the standard elliptic j-invariant. For any r ∈ Q its r-th power is annihilated by the 3-rd order differential operator Hence these are hypergeometric equations.
12.2. Simpson even and odd families. Given an odd positive integer N > 1 consider the hypergeometric series It is an algebraic function of t satisfying P N (f N (t), t) = 0 for a polynomial P N (u, t) ∈ Q[u, t]. This polynomial P N can be given explicitly; the information we need is the shape Hence by Proposition 4 f N (t) r for r ∈ Q satisfies a linear differential equation of order N with singularities only at t = 0, 1, ∞.
In general, the exponents at t = 0, ∞ of the differential equation satisfied by an algebraic function f of this kind can be read-off from its Newton polygon ∆. 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 ∆, say p, q respectively. Let l be the line joining p and q. We can distinguish the top and bottom sides of ∆ as those above and below l respectively.
For each slope κ ∈ Q of a side δ of the Newton polygon consider the sequence where d is the denominator of κ and e is the horizontal width of δ. The exponents at t = 0 for f r are [κ 1 ], [κ 2 ], . . ., where κ 1 , κ 2 , . . . runs over the slopes of the bottom sides. The exponents at t = ∞ 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 f (u, t + 1).
It satisfies an algebraic equation of degree N with Newton polygon the triangle of vertices (0, 0), (1, 0), (N, N ). The exponents are the same as in the case N odd. The multiplicities however are now (1, m − 1, m), (m, m), (1, . . . , 1), where m := N/2. These are the multiplicities of Simpson's even rank case family of rigid local systems [20]. Again we have obtained a geodesic completely contained in the positive components T irr + of the parameter space. For example, for N = 4 we get a geodesic for Goursat G-II up to a twist.
We plot these modulo Z as a function of r (see Figure 2). The first condition (n 1 = 2 in the notation of (3)) for these parameters for generic r to be in the positive components T irr + is that there are two in each of the indicated horizontal strips. This is visible in the plot.
For generic r there should be four δ's in the interval (1/6, 5/6). A plot of these as functions of r modulo Z is given in Figure 3 where this condition is visible.