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danradchenko@gmail.com Max Planck Institute for We study the general properties of certain rank 4 rigid local systems considered by Mathematics, Vivatsgasse 7, Goursat. We analyze when they are irreducible, give an explicit integral description as 53111 Bonn, Germany Dedicated to Don Zagier for his well as the invariant Hermitian form H when it exists. By a computer search, we ﬁnd 65th birthday what we expect are all irreducible such systems all whose solutions are algebraic Full list of author information is functions and give several explicit examples deﬁned over Q. We also exhibit one available at the end of the article 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 F . This was much later extended to hypergeomet- 2 1 ric 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]asour 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 deﬁni- tion 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 P 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 deter- mines the global monodromy representation. In this note, we consider his case II of rank 4 (denoted henceforth by G-II). These are order four linear diﬀerential equations in a variable t with three regular singular points and semisimple local monodromies with © The Author(s) 2018. This 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. 0123456789().,–: volV 38 Page 2 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 2 2 4 eigenvalues of multiplicities 21 , 2 , 1 , respectively (see Sect. 7). We will ﬁx the singular points to be 0, 1, ∞. In modern language, G-II 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 quasi-unipotent.) In this paper, we study the general properties of G-II systems; for example, we ana- lyze when they are irreducible and describe a Hermitian form H invariant under the monodromy group when it exists. This is done in Sect. 4.Asin[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 P are motivic (see [25,p.9]); on the other hand, our construction is explicit. We ﬁnd (see Sect. 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 (see the end of Sect. 4.3 for the general case and Sect. 8 for the case where the ﬁeld of moduli is Q). As in the hypergeometric case, there are inﬁnitely many cases of ﬁnite monodromy G-II local systems which come in families. These families depend linearly on a rational parameter. For G-II, 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 G-II equations whose solutions are algebraic functions and give several explicit examples deﬁned over Q in Sect. 9. In Sect. 6, we show how some G-II 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 G-II 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 ≥ 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 PARI-GP [27] for most of the calculations. 2 Rigid local systems Following the setup and notation of [14], we consider the character variety M where μ is an ordered k-tuple of partitions of a positive integer n. This variety parametrizes represen- tations of π ( \S, ∗)toGL (C) mapping a small oriented loop around s ∈ S to a semisim- 1 n Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 3 of 34 38 s s s ple conjugacy class C whose generic eigenvalues have multiplicities μ = (μ , μ , ...), 1 2 a corresponding partition in μ. Unless otherwise stated in what follows conjugation will always refer to conjugation by GL (C). Here 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 [14]. If non-empty, the variety M is equidimensional of dimension 2 s 2 d := (2g − 2 + k)n − (μ ) + 2. s∈S i≥1 In this paper, we will only consider the case where g = 0 and in detail when k = 3,n = 4 0 2 1 2 ∞ 4 and, taking S ={0, 1, ∞}, the partitions are μ = 21 , μ = 2 , μ = 1 . To be concrete, if g = 0, given conjugacy classes C , ... ,C ⊆ GL (C) and labeling the 1 k n punctures with 1, ... ,k, we are looking for solutions to T ··· T = I,T ∈ C,s = 1, ... ,k, (1) 1 n s s where I is the identity matrix, up to simultaneous conjugation. Given such a represen- tation π ( \ S, ∗), we call the image in :=T , ... ,T ≤ GL (C)the (geometric) 1 1 k n monodromy group. It is well deﬁned up to conjugation. 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 k-tuples of matrices T ∈ C and T ∈ C for s ∈ S s s s −1 satisfying (1), there exists a single U ∈ GL (C) such that T = UT U for all s ∈ S.The n s 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 M consists of a single point. Therefore, it is necessary that the expected dimension d 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 ≥ 3. Indeed, for g = 0,k = 1, the group π ( \ 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 . Such points correspond to apparent singularities in the diﬀerential equation and may hence be safely ignored. We will assume then that the partitions μ have at least two parts. Goursat shows that with the given assumptions k ≤ n + 1[10, top p.114] and hence there are only ﬁnitely many solutions of d = 0 for ﬁxed n.Helists [10, p. 115] the cases of d = 0 for n = 3and n = 4 (see below). It turns out, however, that the condition d = 0 is not suﬃcient as the variety M might μ μ be empty. Crawley-Boevey [5] proved that a necessary and suﬃcient condition for M 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 M is a point. This algorithm ultimately corresponds to Katz’s middle convolution and is simply an explicit implementation of Crawley-Boevey’s criterion. The reader may consult [18] as a general reference for this topic. 38 Page 4 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 It is more convenient to present the multiplicity data μ in the form of a star graph with one central node and k legs (see [5]and also [14]). We illustrate this in our basic case G-II (Goursat’s label II for n = 4). G-II 1 2 4 3 2 1 The partitions can be read by starting at the central node and moving away along a leg. The successive diﬀerences of the respective node values are the parts of the corresponding partition. Nodes with a zero value are not included. For example, in the diagram for G-II given above, there are three legs. The leg to the left has nodes 4, 2, 1 corresponding to the partition (2, 1, 1) since 4 − 2 = 2, 2 − 1 = 1, 1 − 0 = 1, 0 − 0 = 0, .... Similarly, the vertical leg represents the partition (2, 2) and the leg to the right (1, 1, 1, 1). The algorithm proceeds starting from a conﬁguration as above corresponding to an ordered k-tuple μ of partitions of n satisfying d = 0 using the following moves. • A: Replace the value n at the central node by n − n, where n 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 = 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 ≥ μ ≥ ··· of n,wehave nμ ≥ μ . It follows that if d = 0 1 2 1 μ i i s 2 n μ > (2g − 2 + k)n and since also g = 0and n > 0 that μ > (k − 2)n which proves the claim. 2 2 4 For our running example μ = (21 , 2 , 1 ), the algorithm works as follows. Apply A: 1 2 3 3 2 1 Apply B: 1 2 3 2 1 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 5 of 34 38 Apply C: 1 2 3 2 1 3 3 We have arrived at the case μ = (1 , 21, 1 ) that corresponds to the hypergeometric equa- 2 2 2 tion of order3.Itiseasytosee thatanextstage takesusto μ = (1 , 1 , 1 ) 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. This indeed happens for Goursat’s case IV as we verify below. G-IV 1 4 3 2 1 Apply A: G-IV 1 2 3 2 1 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.) 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). 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). G-I 1 2 3 4 3 2 1 G-II 1 2 4 3 2 1 G-III 1 2 4 2 1 38 Page 6 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 G-V 1 4 2 1 2 4 1 G-VI G-VII 4 1 3 Field of deﬁnition and ﬁeld of moduli Given a rigid local system with conjugacy classes C for s ∈ S as in Sect. 2,let q (T)bethe s s characteristic polynomial of any element of C .Let K be the ﬁeld obtained by adjoining to Q the coeﬃcients of all q .Wecall 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 ∈ GL (F) of the required kind, i.e., T ∈ C , may exist. But s n s s 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 Sect. 4.3). Let T ∈ C be a k-tuple of matrices in GL (Q) satisfying (1). It is clear that for σ ∈ s s n Gal(Q/K)the k-tuple T is another solution to (1). Hence by rigidity, there exists X ∈ GL (Q) such that −1 σ X T X = T ,s ∈ S. (2) s σ σ s Again by rigidity, we ﬁnd that there exists a ∈ Q such that σ ,τ X X = a X . σ σ ,τ στ 2 × The map (σ , τ) → a is a 2-cocycle giving a well-deﬁned element ξ ∈ H (Gal(K /K ),K ). σ ,τ 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 ξ is trivial. Note that (2) implies that the trace of any product of T ’s is in the trace ﬁeld K.Thatis, K is indeed the smallest extension of Q containing the traces of all T ∈ . Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 7 of 34 38 4 Explicit solution for the Goursat case II In [10, p. 131], Goursat writes down an explicit solution to (1) for S ={0, 1, ∞} 2 2 2 in the case when T , T ,and T are diagonalizable with spectra 1 a a ,1 b and 0 1 ∞ 1 2 c c c c , respectively (assuming that eigenvalues with diﬀerent labels are distinct and 1 2 3 4 that a a b c c c c = 1). The characteristic polynomials q of T are therefore 1 2 1 2 3 4 s s q (T) = (T − 1) (T − a )(T − a ), 0 1 2 2 2 q (T) = (T − 1) (T − b) , q (T) = (T − c )(T − c )(T − c )(T − c ). ∞ 1 2 3 4 Sometimes it is more convenient to work with the characteristic exponents instead of the eigenvalues. We will use Greek letters to denote them, so that a = exp(2πiα )and j j similarly for β’s and γ ’s. Since the triple (T ,T ,T ) is irreducible, the 1-eigenspaces for T and T must have a 0 1 ∞ 0 1 zero intersection. Goursat then shows that in a suitable basis the matrices T and T are 0 1 given by ⎛ ⎞ ⎛ ⎞ ) B(1 − a ) 10 A(1 − a b 000 1 2 ⎜ ⎟ ⎜ ⎟ 01 C(1 − a ) D(1 − a ) 0 b 00 ⎜ 1 2 ⎟ ⎜ ⎟ T = ,T = . (3) ⎜ ⎟ ⎜ ⎟ 0 1 ⎝ ⎠ ⎝ ⎠ 00 a 0 1 − b 010 00 0 a 01 − b 01 A direct computation shows that for given a , a ,and b, the coeﬃcients of q depend 1 2 ∞ linearly on A, D,and AD − BC. Conversely, the numbers A, D,and AD − BC can be found from q by −1 2 2 −1 (b − 1)(a − 1)(a − a ) a q (a ) − b q (b ) 1 2 1 ∞ ∞ 1 1 A = , b a a a − b −1 2 2 −1 (b − 1)(a − 1)(a − a ) a q (a ) − b q (b ) 2 1 2 ∞ ∞ 2 2 (4) D = , b a a a − b (b − 1) (a − 1)(a − 1) 1 2 2 −1 (AD − BC) = b q (b ). b a a 1 2 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) −1 preserves the shapes of T and T and maps (B, C)to(λ B, λC), hence only the product 0 1 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 and T are 0 1 ⎛ ⎞ ⎛ ⎞ 10 −A −B 00 −10 ⎜ ⎟ ⎜ ⎟ 01 −C −D 00 0 −1 ⎜ ⎟ ⎜ ⎟ Z = ⎜ ⎟ ,Z = ⎜ ⎟ . 0 1 ⎝ 00 1 0 ⎠ ⎝ 10 1 0 ⎠ 00 0 1 01 0 1 38 Page 8 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 One can easily check the following assertions: if C = 0, then the subspace of vectors of the form (∗, 0, ∗, 0) is invariant; if B = 0, then the subspace (0, ∗, 0, ∗) is invariant; if AD − BC = 0, then the subspace spanned by ker(T − I) and the vector (a, c, 0, 0) is invariant; if AD − BC − A − D + 1 = 0, then the subspace spanned by ker(T − bI)and the vector (a − 1,c, 0, 0) is invariant. Conversely, if V is a non-trivial invariant subspace, then considering the various possibilities for V with respect to the eigenspaces of T ,we ﬁnd that one of B, C, AD − BC,or AD − BC − A − D + 1mustvanish. Thus, the representation is irreducible if and only if BC(AD − BC)(AD − BC − A − D + 1) = 0. To get the description in terms of eigenvalues, we use the following factorizations: a a (1 − bc )(1 − bc )(1 − bc )(1 − bc ) 1 2 1 2 3 4 AD − BC = , (1 − b) (1 − a )(1 − a ) 1 2 ba (1 − a bc c ) 1 i j 2 1≤i<j≤4 (5) BC = , 2 2 (a − a ) (1 − b) (1 − a )(1 − a ) 1 2 1 2 (1 − c )(1 − c )(1 − c )(1 − c ) 1 2 3 4 AD − BC − A − D + 1 = . c c c c (1 − b) (1 − a )(1 − a ) 1 2 3 4 1 2 −1 Note that in terms of q this simply becomes q (1) = 0, q (b ) = 0, and ∞ ∞ ∞ −1 −1 w (q )(a b ) = 0, where w (q ) = (T − c c ) is the polynomial whose roots 2 ∞ 2 ∞ i j 1 i<j are products of all pairs of roots of q . This description agrees with the conditions given in [20,p.10]. To summarize, let 2 1 1 T :={(a ,a ,b,c , ... ,c ) | a a b c ··· c = 1}⊆ S × ··· × S 1 2 1 4 1 2 1 4 be the space of eigenvalues (taken in the unit circle). Here is the union of a = 1,a = 1,a = a ,b = 1and c = c for 1 ≤ i < j ≤ 4 guaranteeing that 2 1 2 i j irr (1, 1,a ,a ), (1, 1,b,b), (c , ... ,c ) are the eigenvalues of a G-II system. Deﬁne T as the 1 2 1 4 subset corresponding to irreducible local systems. Then we have irr −1 −1 −1 T = T {q (1)q (b )w (q )(a b ) = 0}. ∞ ∞ 2 ∞ The conditions for irreducibility we found can also be obtained from [5, Thm. 1.5]. Indeed the required decompositions of the real root corresponding to G-II are the follow- ing (and their reﬁnements). Let i , ... ,i be any re-ordering of 1, ... , 4. 1 4 i) q (1) = 0 1 a a 1 1 2 1 bb 1,a a b c c c = 1,c = 1. 1 2 i i i i 1 2 3 4 c c c c i i i i 1 2 3 4 −1 ii) q (b ) = 0 1 a a 1 1 2 11 b b,a a bc c c = 1,bc = 1. 1 2 i i i i 1 2 3 4 c c c c i i i i 1 2 3 4 −1 −1 iii) w (q )(a b ) = 0 2 ∞ 1 a 1 a 1 2 1 b 1 b,a bc c = 1,a bc c = 1. 1 i i 2 i i 1 2 3 4 c c c c i i i i 1 2 3 4 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 9 of 34 38 4.2 Invariant Hermitian form Let T , ... ,T ∈ GL (C) correspond to an irreducible local system. Assume that there 1 k n exists a nonzero Hermitian form H on C invariant under the group =T , ... ,T , i.e., 1 k T HT = H, s = 1, ... ,k. (6) s s Since ker(H) is invariant under all T by irreducibility, we get that any such H must be ∗ −1 non-degenerate. This implies, in particular, that (T ) and T are conjugate. Therefore, s s −1 the sets of eigenvalues of T are invariant under the map z → z¯ . This is certainly the case if the eigenvalues are in the unit circle. −1 On the other hand, if the eigenvalues of T are invariant under the map z → z for all ∗ −1 s then the (T ) 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. irr The set T 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 <α and γ <γ <γ <γ (recall that 1 2 1 2 3 4 exp(2πiα ) = a and so on). Then we ﬁnd that there is a unique connected component j j 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 } and {B } are two collections of matrices, then we can k k easily test if they are simultaneously conjugate by solving the system A X − XB = 0. In k k our computations with monodromy groups, we often rely on this observation. We can compute the invariant form explicitly starting from (3). Equation (4)implies 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 ⎛ ⎞ C(1 − DE) BCE C(1 − D) BC ⎜ ⎟ BCE B(1 − AE) BC B(1 − A) ⎜ ⎟ H = (AD − BC) , (7) ⎜ ⎟ ⎝ ⎠ C(1 − D) BC C(1 − D) BC BC B(1 − A) BC B(1 − A) where E = (A + D − 1)/(AD − BC). The determinant of H is 2 3 3 (BC) (AD − BC − A − D + 1) (AD − BC) . irr Using (7), we can easily describe T in terms of the parameters (A, D, t) where t = BC. If we look at the connected components of the set R V , where V ={(A, D, t) | t(t − AD)(t − (1 − A)(1 − D)) = 0}, and compute the signature in each case, we ﬁnd that H is positive deﬁnite if and only if 0 < A, D < 1, 0 < BC < AD, 0 < BC < (1 − A)(1 − D). 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 be the open arc in S with end −1 points 0, and b (any of the two possibilities), and let I be the arc with end points 2 38 Page 10 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 −1 −1 −1 −1 −1 (b a ,b a ), where among the two arcs we pick the one that contains the point b . 1 2 Proposition 2 The invariant Hermitian form H is deﬁnite if and only if for some labeling c , ... ,c of the eigenvalues of T we have 1 4 ∞ (i) c ,c ∈ I ,c ,c ∈ / I , 1 2 1 3 4 1 (ii) c c ,c c ,c c ,c c ∈ I ,c c ,c c ∈ / I . 1 2 3 4 1 3 2 4 2 1 4 2 3 2 4.3 Integrality The matrices given by Goursat (3), when expressed in terms of the eigenvalues, have non- trivial 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). 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 as follows. ⎛ ⎞ ⎛ ⎞ −1 −1 σ 00 σ a 000 −τ 1 4 2 1 ⎜ −1 −1 ⎟ ⎜ ⎟ σ (a + 1) σ 1 σ σ a 100 τ ⎜ 2 1 1 1 ⎟ ⎜ 3 ⎟ 2 1 T = ,T = , ⎜ ⎟ ⎜ ⎟ 1 ∞ −1 ⎝ ⎠ ⎝ ⎠ −σ σ a −σ 01 + a 010 −τ 1 2 1 2 2 −σ a 00 0 001 τ 1 1 where σ = e (b ,b ), τ = e (c ,c ,c ,c ) are the elementary symmetric functions. i i 1 2 i i 1 2 3 4 With these, using that a a σ τ = 1, 1 2 4 obtained by taking determinants in T T T = I ,weget 0 1 ∞ 4 ⎛ ⎞ −1 (a + 1) 0 −σ a (σ τ − τ ) 1 2 1 4 3 ⎜ −1 −1 ⎟ −σ a 1 σ σ a (τ − σ τ ) − σ ⎜ 1 1 1 2 2 2 4 ⎟ 2 2 T = . ⎜ ⎟ −1 ⎝ ⎠ σ a00 −a τ + σ σ 2 1 2 1 1 00 0 a The trace ﬁeld is generically given by K = Q(σ , σ , τ , ... , τ ) and we see that we can 1 2 1 4 always take as ﬁeld of deﬁnition the quadratic extension F := K (a ). Note that we also have tr(T ) = 2 + a + a ∈ K . Hence a and a are conjugate over K . 0 1 2 1 2 In fact, the local monodromies are deﬁnable over the ring R[a ], where R := −1 −1 Z[σ , σ , τ , ... , τ , σ , τ ] and hence the group they generate as well. The traces 1 2 1 4 2 4 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 Intro- duction) the characteristic polynomials q ,q ,q will have only roots of unity as roots. In 0 1 ∞ this case, K is a cyclotomic ﬁeld. We conclude that the monodromy can be conjugated to lie in GL (O ), where O is the ring of integers of F = K (a ). This is consistent with the 4 F F 1 rigid local system being motivic. 2 2 2 2 4 3 3 For example, consider q = (x − 1) (x + 1),q = (x − 1) ,q = x + (ζ − ζ )x − 0 1 ∞ 12 ζ x + 1, where ζ is a primitive 12-root of unity. This corresponds to row #3 in Table 2. 12 12 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 11 of 34 38 Then our choice gives ⎛ ⎞ ⎛ ⎞ 3 2 3 ζ + 101 ζ − 1 000 ζ 12 12 12 ⎜ ⎟ ⎜ ⎟ 3 3 010 −ζ + 1 ζ + 101 0 ⎜ ⎟ ⎜ ⎟ 12 12 T := ,T := , ⎜ ⎟ ⎜ ⎟ 0 1 3 2 3 ⎝ ⎠ ⎝ ⎠ −ζ 00 ζ 010 −ζ + 1 12 12 12 3 3 000 −ζ −ζ 00 0 12 12 and ⎛ ⎞ 000 −1 ⎜ ⎟ 100 ζ ⎜ 12 ⎟ T := ⎜ ⎟ . ⎝ 010 0 ⎠ 001 −ζ + ζ These matrices generate a group of order 103680, which is a non-split central extension by C of the simple group . We see here a phenomenon that occurs frequently in 4 25920 our examples. The quotient /Z() has no irreducible representation of degree 4 (the smallest non-trivial irreducible representation is of order 5), whereas a central extension, namely ,does. It follows from the above discussion that for G-II cases the cocycle obstruction of Sect. 3 is generically of order dividing 2 = [F : K ]. We can easily compute the corresponding matrix X for σ the generator of Gal(F /K ) as in Sect. 3. The problem is linear: we solve T X = X T generically, where σ (a ) = a .Weﬁnd s σ σ 1 2 X X = μI , (8) σ 4 3 −1 −1 × where μ =−(a σ ) w (q ) a σ ∈ K . Recall that w (q ):= (T − c c ). 1 2 2 ∞ 2 ∞ i j 1 2 i<j The cocycle can be represented by a quaternion algebra. Explicitly, this is the quaternion D,μ algebra , where D = disc(F)and μ is as above. 5 Finite monodromy We would like to describe all cases of G-II 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 ⊆ GL (C), an identiﬁcation of A and the quotient of /A using standard notation (A denotes a maximal abelian normal subgroup of ), the order of the center of and whether acts primitively or not. By a theorem of Jordan, there are ﬁnitely many possibilities for the quotient /A.The ﬁnite groups acting in four dimensions were classiﬁed by Blichfeldt (see [11] for a modern description). The group denoted by is a simple group. 25920 38 Page 12 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Table 1 α , α = 1/3, 2/3 1 2 γ || /AA |Z()| Impr 11/8, 3/8, 5/8, 7/848 S C 2 ∗ 4 2 21/5, 2/5, 3/5, 4/560 A 11 31/10, 3/10, 7/10, 9/10 120 A C 2 5 2 41/12, 5/12, 7/12, 11/12 144 C × A C 2 ∗ 2 4 6 51/20, 9/20, 13/20, 17/20 240 A C 4 5 4 62/9, 1/3, 5/9, 8/9 324 A C 1 ∗ 71/24, 7/24, 17/24, 23/24 576 S × A C 2 4 4 2 81/28, 9/28, 3/4, 25/28 672 PSL (F ) C 4 2 7 4 91/20, 9/20, 11/20, 19/20 720 C × A C 2 ∗ 2 5 6 10 1/15, 4/15, 11/15, 14/15 1440 A × A C 2 5 4 2 11 1/30, 11/30, 19/30, 29/30 1440 A × A C 2 5 4 2 12 1/40, 9/40, 31/40, 39/40 2880 A × S C 2 5 4 2 Table 2 α , α = 1/4, 3/4 1 2 γ || /AA |Z()| Impr 11/12, 5/12, 7/12, 11/12 192 C × S C 2 ∗ 2 4 4 21/20, 9/20, 13/20, 17/20 640 C D C 4 5 4 31/36, 13/36, 25/36, 11/12 103680 C 4 25920 4 Table 3 α , α = 1/5, 4/5 1 2 γ || /AA |Z()| Impr 11/12, 5/12, 7/12, 11/12 1200 C × A C 2 ∗ 2 5 10 22/15, 7/15, 8/15, 13/15 7200 A × A C 2 5 5 2 31/20, 9/20, 11/20, 19/20 1200 C × A C 2 ∗ 2 5 10 41/30, 11/30, 19/30, 29/30 7200 A × A C 2 5 5 2 We should note that we can always twist the local monodromies by multiplying by scalars matrices so that the resulting triple is in SL (C). If the group acts primitively, the normal subgroup A consists of scalars. It follows that there are ﬁnitely many possible primitive up to twisting; we will see in Sect. 12 that this is not the case for imprimitive groups. 5.2 Special case We start by discussing a special, simpler case. Assume that the characteristic polynomials q ,q ,q of the local monodromies at the respective singularities have real coeﬃcients 0 1 ∞ 2 2 and that q = (T −1) (T +1) .Let γ ∈ (0, 1) for i = 1, ... , 4 be the exponents of the roots 1 i of q (so that c = exp(2πiγ )) and similarly let α ∈ (0, 1/2) be such that the exponents ∞ j j 1 of q are 0, 0, α , 1 − α . 0 1 1 A special case of Proposition 2 reduces in this case to the following. Let δ , ... , δ be 1 6 representatives in (0, 1) (with multiplicities) of the exponents γ + γ for i < j. Deﬁne i j n as the number of γ in the interval (0, 1/2) and n the number of δ (counting with 1 i 2 i multiplicities) in the interval (1/2 − α , 1/2 + α ). 1 1 Proposition 3 With the above assumptions and notations the invariant Hermitian form H is deﬁnite if and only if (n ,n ) = (2, 4). 1 2 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 13 of 34 38 Table 4 General case β, α , α γ || /AA |Z()| Impr 1 2 11/2, 1/2, 1/31/8, 11/24, 5/8, 23/24 4608 (A × A ) C C 4 ∗ 4 4 2 21/2, 1/2, 1/35/48, 23/48, 29/48, 47/48 41472 (A × A ) D C 6 ∗ 4 4 4 31/2, 1/2, 1/311/120, 59/120, 71/120, 119/120 1036800 (A × A ) C C 12 ∗ 5 5 2 41/2, 1/2, 1/47/48, 23/48, 31/48, 47/48 6144 C D C · C 8 ∗ 6 4 8 51/2, 1/2, 1/57/40, 19/40, 27/40, 39/40 2880000 (A × A ) C C 20 ∗ 5 5 2 61/2, 1/2, 1/519/120, 59/120, 79/120, 119/120 2880000 (A × A ) C C 20 ∗ 5 5 2 71/2, 1/2, 1/65/36, 17/36, 29/36, 11/12 311040 C 12 25920 12 81/2, 1/2, 1/611/60, 23/60, 47/60, 59/60 311040 C 12 25920 12 91/2, 1/3, 1/47/24, 5/12, 19/24, 11/12 165888 (A × A ) D C 12 ∗ 4 4 4 10 1/2, 1/3, 1/411/48, 23/48, 35/48, 47/48 165888 (A × A ) D C 12 ∗ 4 4 4 11 1/2, 1/3, 1/54/15, 7/15, 23/30, 29/30 6480000 (A × A ) C C 30 ∗ 5 5 2 12 1/2, 1/3, 1/517/60, 9/20, 47/60, 19/20 6480000 (A × A ) C C 30 ∗ 5 5 2 13 1/2, 1/3, 1/519/60, 5/12, 49/60, 11/12 6480000 (A × A ) C C 30 ∗ 5 5 2 14 1/2, 1/3, 1/529/120, 59/120, 89/120, 119/120 6480000 (A × A ) C C 30 ∗ 5 5 2 15 1/2, 1/5, 2/54/15, 13/30, 23/30, 14/15 6000 S C · C 10 ∗ 5 2 16 1/2, 1/5, 2/59/40, 19/40, 29/40, 39/40 6000 S C · C 10 ∗ 5 2 17 1/3, 1/2, 5/61/18, 7/18, 13/18, 5/6 155520 C 6 25920 6 18 1/3, 1/2, 1/65/18, 11/18, 5/6, 17/18 155520 C 6 25920 6 19 1/3, 1/2, 1/611/30, 17/30, 23/30, 29/30 155520 C 6 25920 6 20 1/3, 1/3, 2/31/12, 11/24, 5/6, 23/24 69120 C .A C 12 6 12 21 1/3, 1/3, 2/32/15, 8/15, 11/15, 14/15 2160 A C 6 6 6 22 1/3, 1/3, 2/35/24, 11/24, 17/24, 23/24 2160 A C 6 6 6 23 1/3, 1/3, 2/35/42, 17/42, 5/6, 41/42 15120 A C 6 7 6 24 1/3, 1/3, 2/311/60, 23/60, 47/60, 59/60 69120 C .A C 12 6 12 To illustrate the situation, here is a picture with the position of the various roots on the unit circle in the case α , α = 1/3, 2/3and γ = (1/28, 9/28, 3/4, 25/28). Hence 1 2 δ = (1/14, 3/14, 5/14, 9/14, 11/14, 13/14). n = 2 1 0 4 38 Page 14 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 14 1 n = 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 Here we consider the general case (up to twisting) where the exponents are t exponents 0 0, 0, α , α 1 2 1 0, 0, β, β ∞ γ , γ , γ , γ 1 2 3 4 The ﬁnite monodromy cases found are listed in Table 4. 6 Coxeter groups To ﬁnd explicit realizations of ﬁnite monodromy groups of G-II type, we may start with a ﬁnite group in GL (C) and attempt to build a G-II rigid local system by producing three appropriate elements T ,T ,T . For example, we can take a ﬁnite complex reﬂection 0 1 ∞ group W in rank 4, hence one of the Weyl groups A ,B ,F or the non-crystallographic 4 4 4 case H .Since T should have distinct eigenvalues diﬀerent from 1, we could start by 4 ∞ taking T to be a Coxeter element. Similarly, we can take T to be the product of two ∞ 1 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 above procedure in one example in the case H here and give several further examples deﬁned over Q in Sect. 9. The Dynkin diagram is H : where we have circled the two chosen simple reﬂections. We take T := s s s s ,T := ∞ 1 2 3 4 1 s s and T so that T T T = 1. 1 3 0 0 1 ∞ With the help of MAGMA (see the actual calculations below), we ﬁnd that ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 ττ 0 −10 00 −1 −τ −τ −τ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 00 −10 τ 110 ττττ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T = ⎜ ⎟ ,T = ⎜ ⎟ ,T = ⎜ ⎟ , 0 1 ∞ ⎝ 0011 ⎠ ⎝ 00 −10 ⎠ ⎝ 0100 ⎠ 0 −1 −1 −1 00 11 0010 where τ − τ − 1 = 0. Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 15 of 34 38 Taking the embedding into R where τ = (1 + 5)/2 = 1.618033988 ··· is the golden ratio, the exponents of the local monodromies are: T :(0, 0, 1/3, 2/3),T :(0, 0, 1/2, 1/2),T :(1/30, 11/30, 19/30, 29/30) 0 1 ∞ and hence this example corresponds to row #11 in the above table for α , α = 1/3, 2/3. 1 2 This shows, in particular, that here the monodromy representation can be realized over Z[τ], the ring of integers of the trace ﬁeld K := Q( 5). This is consistent with our discussion in Sect. 4.3. > W<s1,s2,s3,s4>:=CoxeterGroup(GrpMat,"H4"); > K<a>:=BaseRing(W); >K; Number Field with defining polynomial xˆ2-x-1 over the Rational Field > R<x>:=PolynomialRing(K); > Tinf:=s1*s2*s3*s4; > T1:=s1*s3; > T0:=Tinfˆ(-1)*T1ˆ(-1); > CharacteristicPolynomial(Tinf); xˆ4 + (-a + 1)*xˆ3 + (-a + 1)*xˆ2 + (-a + 1)*x + 1 > CharacteristicPolynomial(T1); xˆ4 - 2*xˆ2 + 1 > CharacteristicPolynomial(T0); xˆ4-xˆ3-x+1 > T0; [ 1aa0] [0 0-1 0] [ 0011] [0-1-1-1] > T1; [-1000] [ a110] [0 0-1 0] [ 0011] > Tinf; [-1 -a -a -a] [ aaaa] [ 0100] [ 0010] > G:=sub<W|[T1,Tinf]>; > #G; 7 Diﬀerential equation Goursat [10, §10] computes explicitly an order four linear diﬀerential equation of type G-II with given local monodromies. Let the exponents of these local monodromies be 0:(0, 1, 1 − α , 1 − α ), 1:(0, 1, β, β + 1), ∞ :(γ , γ , γ , γ ). 1 2 1 2 3 4 These are related by the equation β = (1 + e (α) − e (γ )), α := (α , α ), γ := (γ , ... , γ ), 1 1 1 2 1 4 where e (x) denotes the elementary symmetric functions of the quantities x = (x ,x , ...). n 1 2 38 Page 16 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 The shape of the exponents forces the diﬀerential equation to be of the following form 4 3 2 d d d 2 2 2 x (x − 1) + (Ax − B)x(x − 1) + (Cx − Dx + E) 4 3 2 dx dx dx + (Fx − G) + H = 0. dx for certain constants A, B, ... ,H. Imposing further that there are no logarithmic solu- tions at x = 1 completely determines these constants. Following Goursat, we obtain the following values (the expression for G he gives is not quite right and is here corrected.) A = 6 + e (γ ), B = 3 + e (α), C = 7 + 3e (γ ) + e (γ ), 1 2 D = E + C − (β − 1)(β − 2), E = 1 + e (α) + e (α), 1 2 F = 1 + e (γ ) + e (γ ) + e (γ ), 1 2 3 G = F + 2(β − 1)(β − 2)(β − 3) + (β − 1)(β − 2)(2A − B) + (β − 1)(2C − D), H = e (γ ). As worked out also by Goursat [10, (18)], the coeﬃcients a of a power series solution a x to the diﬀerential equation satisfy the second-order recursion n≥0 C a = C a + C a , (9) 0 n 1 n−1 2 n−2 where C := (n + 1)(n + 2)[n(n − 1) + Bn + E], C := (n + 1)[2n(n − 1)(n − 2) + (A + B)n(n − 1) + Dn + G], C :=−[n(n − 1)(n − 2)(n − 3) + An(n − 1)(n − 2) + Cn(n − 1) + Fn + H]. For example, if we take α = (1/4, 3/4) and γ = (1/5, 2/5, 3/5, −1/5) (so that β = 1/2) we obtain the diﬀerential equation 4 3 d d 2 2 x (x − 1) + x(x − 1)(7x − 4) 4 3 dx dx d d +(51/5x − 931/80x + 35/16) + (54/25x − 1223/800) − 6/625, (10) dx dx which indeed has local exponents 0:(0, 1, 1/4, 3/4), 1:(0, 1/2, 1, 3/2), ∞ :(−1/5, 1/5, 2/5, 3/5). The ﬁrst few coeﬃcients of the power series expansion of a basis of holomorphic solutions to the diﬀerential equation at x = 0 are as follows: 2 3 4 φ = 1 + 48/21875x + 28088/18046875x + 6589643/5865234375x 5 6 + 57582020413/67659667968750x + O(x ), (11) 2 3 4 φ = x + 1223/3500x + 1096811/5775000x + 370276451/3003000000x 5 6 + 15278570717561/173208750000000x + O(x ). Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 17 of 34 38 Table 5 Signature (2, 2) α , α β , β γ , γ , γ , γ D μ 1 2 1 2 1 2 3 4 11/3, 2/30, 1/21/4, 1/3, 2/3, 3/4 −3 −2 21/3, 2/30, 1/21/6, 1/4, 3/4, 5/6 −3 −2 31/3, 2/31/3, 2/31/6, 1/4, 3/4, 5/6 −3 −2 41/3, 2/31/3, 2/31/10, 3/10, 7/10, 9/10 −3 −1 51/3, 2/31/6, 5/61/4, 1/3, 2/3, 3/4 −3 −2 61/3, 2/31/6, 5/61/5, 2/5, 3/5, 4/5 −3 −1 71/4, 3/40, 1/21/4, 1/3, 2/3, 3/4 −4 −3 81/4, 3/40, 1/21/6, 1/3, 2/3, 5/6 −4 −1 91/4, 3/40, 1/21/6, 1/4, 3/4, 5/6 −4 −3 10 1/4, 3/40, 1/21/5, 2/5, 3/5, 4/5 −4 −1 11 1/4, 3/40, 1/21/10, 3/10, 7/10, 9/10 −4 −1 12 1/4, 3/41/3, 2/31/6, 1/4, 3/4, 5/6 −4 −3 13 1/4, 3/41/3, 2/31/10, 3/10, 7/10, 9/10 −4 −1 14 1/4, 3/41/3, 2/31/12, 5/12, 7/12, 11/12 −41 15 1/4, 3/41/4, 3/41/12, 5/12, 7/12, 11/12 −41 16 1/4, 3/41/6, 5/61/4, 1/3, 2/3, 3/4 −4 −3 17 1/4, 3/41/6, 5/61/5, 2/5, 3/5, 4/5 −4 −1 18 1/4, 3/41/6, 5/61/12, 5/12, 7/12, 11/12 −41 19 1/6, 5/60, 1/21/4, 1/3, 2/3, 3/4 −3 −2 20 1/6, 5/60, 1/21/6, 1/3, 2/3, 5/6 −3 −2 21 1/6, 5/60, 1/21/6, 1/4, 3/4, 5/6 −3 −2 22 1/6, 5/60, 1/21/5, 2/5, 3/5, 4/5 −3 −1 23 1/6, 5/60, 1/21/8, 3/8, 5/8, 7/8 −3 −1 24 1/6, 5/60, 1/21/10, 3/10, 7/10, 9/10 −3 −1 25 1/6, 5/61/3, 2/31/6, 1/4, 3/4, 5/6 −3 −2 26 1/6, 5/61/3, 2/31/5, 2/5, 3/5, 4/5 −31 27 1/6, 5/61/3, 2/31/8, 3/8, 5/8, 7/8 −31 28 1/6, 5/61/3, 2/31/10, 3/10, 7/10, 9/10 −31 29 1/6, 5/61/3, 2/31/12, 5/12, 7/12, 11/12 −32 30 1/6, 5/61/4, 3/41/5, 2/5, 3/5, 4/5 −31 31 1/6, 5/61/4, 3/41/8, 3/8, 5/8, 7/8 −31 32 1/6, 5/61/4, 3/41/10, 3/10, 7/10, 9/10 −31 33 1/6, 5/61/4, 3/41/12, 5/12, 7/12, 11/12 −32 34 1/6, 5/61/6, 5/61/4, 1/3, 2/3, 3/4 −3 −2 35 1/6, 5/61/6, 5/61/5, 2/5, 3/5, 4/5 −31 36 1/6, 5/61/6, 5/61/8, 3/8, 5/8, 7/8 −31 37 1/6, 5/61/6, 5/61/10, 3/10, 7/10, 9/10 −31 38 1/6, 5/61/6, 5/61/12, 5/12, 7/12, 11/12 −32 As expected from a motivic situation the denominators of the coeﬃcients appear to grow only exponentially, rather than what could be expected generically from solutions to recursion (9). 8 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. We list the results in Tables 5 and 6 according to the signature of the respective Hermitian form. 38 Page 18 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Table 6 Signature (4, 0) α , α β , β γ , γ , γ , γ D μ 1 2 1 2 1 2 3 4 11/3, 2/30, 1/21/5, 2/5, 3/5, 4/5 −31 21/3, 2/30, 1/21/8, 3/8, 5/8, 7/8 −31 31/3, 2/30, 1/21/10, 3/10, 7/10, 9/10 −31 41/3, 2/30, 1/21/12, 5/12, 7/12, 11/12 −32 51/3, 2/31/3, 2/31/5, 2/5, 3/5, 4/5 −3 −1 61/3, 2/31/3, 2/31/8, 3/8, 5/8, 7/8 −3 −1 71/3, 2/31/4, 3/41/6, 1/3, 2/3, 5/6 −3 −2 81/3, 2/31/4, 3/41/5, 2/5, 3/5, 4/5 −3 −1 91/3, 2/31/4, 3/41/8, 3/8, 5/8, 7/8 −3 −1 10 1/3, 2/31/4, 3/41/10, 3/10, 7/10, 9/10 −3 −1 11 1/3, 2/31/6, 5/61/8, 3/8, 5/8, 7/8 −3 −1 12 1/3, 2/31/6, 5/61/10, 3/10, 7/10, 9/10 −3 −1 13 1/4, 3/40, 1/21/12, 5/12, 7/12, 11/12 −41 14 1/4, 3/41/3, 2/31/5, 2/5, 3/5, 4/5 −4 −1 15 1/4, 3/41/4, 3/41/6, 1/3, 2/3, 5/6 −4 −1 16 1/4, 3/41/4, 3/41/5, 2/5, 3/5, 4/5 −4 −1 17 1/4, 3/41/4, 3/41/10, 3/10, 7/10, 9/10 −4 −1 18 1/4, 3/41/6, 5/61/10, 3/10, 7/10, 9/10 −4 −1 Table 7 Quaternion algebras D\μ −3 −2 −11 2 −3[3, ∞][2, ∞][3, ∞][][2, 3] −4[3, ∞][2, ∞][2, ∞][] Recall that the obstruction for the realizability of the monodromy group over the ﬁeld D,μ of moduli is given by the quaternion algebra , where D = disc(F), with F = Q(a ), and μ is the number computed in (8). There are only four diﬀerent quaternion algebras over Q that appear depending on α , α . To give these algebras is enough to give the list 1 2 [p , ... ,p ] of ramiﬁed primes. These are: [ ] (i.e., the matrix algebra), [2, ∞], [3, ∞]or 1 2r [2, 3] (see Table 7). 9 Finite monodromy K = Q As shown in Table 6, there are only four cases of ﬁnite monodromy with ﬁeld of moduli Q that can be realized over R (rows #1, #2, #4, and #14; note that #3 is a twist of #1). Three cases are actually deﬁnable over 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 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) We can ﬁnd this case as a subgroup of S viewed as the Coxeter group of the root system A . A : The monodromy group is isomorphic to A , the alternating group in ﬁve letters, acting in its standard representation. Here is a calculation using MAGMA. Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 19 of 34 38 > W<s1,s2,s3,s4>:=CoxeterGroup(GrpMat,"A4"); > K:=BaseRing(W); > R<x>:=PolynomialRing(K); > Tinf:=s1*s2*s3*s4; > T1:=s1*s3; > T0:=Tinfˆ(-1)*T1ˆ(-1); > CharacteristicPolynomial(T0); xˆ4-xˆ3-x+1 > CharacteristicPolynomial(T1); xˆ4 - 2*xˆ2 + 1 > CharacteristicPolynomial(Tinf); xˆ4+xˆ3+xˆ2+x+1 > G:=sub<W|[T0,T1,Tinf]>; > IsIsomorphic(G,AlternatingGroup(5)); true Homomorphism of MatrixGroup(4, Rational Field) of order 2ˆ2*3*5 into GrpPerm: $, Degree 5, Order 2ˆ2*3*5 induced by [ 1110] [0 0-1 0] [ 0011] [ 0 -1 -1 -1] |--> (2, 5, 3) [-1000] [ 1110] [0 0-1 0] [ 0011] |--> (1, 2)(4, 5) [-1 -1 -1 -1] [ 1000] [ 0100] [ 0010] |--> (1, 3, 5, 4, 2) Choosing the parameters in Goursat’s diﬀerential equation (Sect. 7)as (α , α ) = (1/3, 2/3); β = 1/2; (γ , γ , γ , γ ) = (1/5, 2/5, −2/5, 4/5) 1 2 1 2 3 4 we obtain 7, 4, 10, 413/36, 20/9, 46/25, 2387/1800, −16/625 for the eight constants A, B, ... ,H. Then all holomorphic solutions at x = 0 have power series expansion, since they represent algebraic functions of x, with integral coeﬃcients, up to the power of some constant N. (The minimal such N is called the Eisenstein constant of the algebraic function; in this example it seems to only involve the primes 2, 5 and 11.) The holomorphic solution to equation holomorphic at x = 0 and starting as 2 3 4 y := 1 − 387/1300 x − 172773/2080000 x − 141382989/3328000000 x + O(x ) satisﬁes an algebraic equation of degree 10 over Q(x). (The series can be computed using the explicit form of the diﬀerential equation given by Goursat, see Sect. 7.) The solution over K := Q( −15) that starts as 2 3 1 − (123/475 + 33/1900 ω)x − (271713/3800000 + 78771/15200000 ω)x + O(x ) where ω − ω + 4 = 0 is a generator of the ring of integers of K on the other hand, satisﬁes the following degree ﬁve equation 5 3 2 P(x, y):= y + a (x)y + a (x)y + a (x)y + a (x) = 0, 3 2 1 0 38 Page 20 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 where a := 605/8664 − 715/2888 ω, a :=−1189825/2963088 + 70525/329232 ω, a := (298150/390963 − 11050/130321 ω)x − 518989705/900778752 + 19234735/300259584 ω, a :=−(453252/2476099 + 151020/2476099 ω)x + (3663787/14856594 + 406915/4952198 ω)x − (82982887/900778752 + 9216415/300259584 ω). 9.2 (1/4, 3/4), (0, 1/2), (1/8, 3/8, 5/8, 7/8) We can ﬁnd this case as a subgroup of the Coxeter group of the root system B . B : The monodromy group is isomorphic to GL (F ), of order 48, in its unique faithful 2 3 irreducible representation of dimension four. Here is a calculation using MAGMA. > W<s1,s2,s3,s4>:=CoxeterGroup(GrpMat,"B4"); > K:=BaseRing(W); > R<x>:=PolynomialRing(K); > Tinf:=s1*s2*s3*s4; > T1:=s2*s4; > T0:=Tinfˆ(-1)*T1ˆ(-1); > CharacteristicPolynomial(T0); xˆ4-xˆ3-x+1 > CharacteristicPolynomial(T1); xˆ4 - 2*xˆ2 + 1 > CharacteristicPolynomial(Tinf); xˆ4+1 > T0; [0-1 0 0] [ 0112] [ 1110] [-1 -1 -1 -1] > T1; [ 1100] [0-1 0 0] [ 0112] [ 000 -1] > Tinf; [-1 -1 -1 -2] [ 1000] [ 0100] [ 0011] > G:=sub<W|[T0,T1,Tinf]>; > IsIsomorphic(G,GL(2,GF(3))); true Mapping from: GrpMat: G to GL(2, GF(3)) Composition of Mapping from: GrpMat: G to GrpPC and Mapping from: GrpPC to GrpPC and Mapping from: GrpPC to GL(2, GF(3)) Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 21 of 34 38 We can change basis so that T is the companion matrix of = T + 1. We obtain ∞ 8 the following triple: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0110 −1 −1 −1 −1 000 −1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ −10 −10 000 −1 100 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T = ⎜ ⎟ ,T = ⎜ ⎟ ,T = ⎜ ⎟ 0 1 ∞ ⎝ 0 −100 ⎠ ⎝ 01 01 ⎠ ⎝ 010 1 ⎠ 0101 −1001 001 0 Choosing the parameters in Goursat’s diﬀerential equation below (Sect. 7)as (α , α ) = (1/4, 3/4); β = 1/2; (γ , γ , γ , γ ) = (1/8, 3/8, −3/8, 7/8) 1 2 1 2 3 4 we obtain 7, 4, 319/32, 3295/288, 20/9, 117/64, 383/288, −63/4096 for the eight constants A, B, ... ,H. Then all holomorphic solutions at x = 0 have power series expansion with integral coeﬃcients up to powers of 2 and 5. The holomorphic solution to equation holomorphic at x = 0 and starting as √ √ 2 3 1 + (5/256 −8 − 29/128)x + (383/65536 −8 − 527/8192)x + O(x ) satisﬁes the degree eight equation 8 6 4 2 P(x, y) = y + a (x)y + a (x)y + a (x)y + a (x), 6 4 2 0 where a := 230/729 −8 − 400/729, a := (1048/19683 −8 + 19984/19683)x − (351670/1594323 −8 + 1034482/1594323), a := (4842880/43046721 −8 − 10078688/43046721)x, + (−1015591450/10460353203 −8 + 1684358888/10460353203), a :=−(27028768/1162261467 −8 + 3467632/1162261467)x + (172219360/3486784401 −8 + 238769752/3486784401)x − (296048878/10460353203 −8 + 1067187679/10460353203)x + (22649710/10460353203 −8 + 382087111/10460353203). 2 2 In addition, let φ := 1+0·x +O(x ), φ := 0·1+x +O(x ) be a basis of the holomorphic 0 1 solutions to the diﬀerential equation at x = 0 and deﬁne 2 2 ψ := φ − 891/16384φ . 0 1 Then ψ satisﬁes a degree four equation; more precisely, if ξ is the hypergeometric series satisfying the trinomial equation 4 3 ξ − 4ξ + 27x = 0, 3 2 then ψ =−4/135ξ + 4/45ξ + 32/45ξ − 37/27. 9.3 (1/4, 3/4), (0, 1/2), (1/12, 5/12, 7/12, 11/12) We can ﬁnd this case as a subgroup of the Coxeter group of the root system F . F : 4 38 Page 22 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 The monodromy group is isomorphic to U (F ) C , of order 192, in one of its irre- 2 3 2 ducible faithful representations of dimension four. This group is labeled (192, 988) in the SmallGroup database. Here is a calculation using MAGMA. > W<s1,s2,s3,s4>:=CoxeterGroup(GrpMat,"F4"); > K:=BaseRing(W); > R<x>:=PolynomialRing(K); > Tinf:=s1*s2*s3*s4; > T1:=s1*s3; > T0:=Tinfˆ(-1)*T1ˆ(-1); > CharacteristicPolynomial(T0); xˆ4 - 2*xˆ3 + 2*xˆ2 - 2*x + 1 > CharacteristicPolynomial(T1); xˆ4 - 2*xˆ2 + 1 > CharacteristicPolynomial(Tinf); xˆ4-xˆ2+1 > T0; [ 1120] [ 0100] [ 0011] [0-1-2-1] > T1; [-1000] [ 1120] [0 0-1 0] [ 0011] > Tinf; [-1 -1 -2 -2] [ 1000] [ 0111] [ 0010] > G:=sub<W|[T0,T1,Tinf]>; > IdentifyGroup(G); <192, 988> 4 2 Conjugating so that T is the companion matrix of = T − T + 1weobtainthe ∞ 12 following triple ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 100 −1 0111 000 −1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0101 10 0 −1 100 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T = ,T = ,T = . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 1 ∞ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0011 00 −10 010 1 0 −1 −1 −1 00 1 1 001 0 The permutation representation of the smallest degree for this monodromy group is of degree 24. Hence some solution to the diﬀerential equation satisﬁes a degree 24 equation with coeﬃcients in some number ﬁeld but we have not attempted to ﬁnd it. 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 = Q( 5), the obstruction cocycle ξ 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 non-crystallographic W (H ) of order 14400. Note, however, that in this 4 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 23 of 34 38 case T is not a Coxeter element; we still take T as a product of two commuting simple ∞ 1 reﬂections. Here are some of the computations in MAGMA. > W<s1,s2,s3,s4>:=CoxeterGroup(GrpMat,"H4"); > K<a>:=BaseRing(W); >K; Number Field with defining polynomial xˆ2-x-1 over the Rational Field > R<x>:=PolynomialRing(K); > CC:=ConjugacyClasses(W); > [Order(g[3]): g in CC]; [1,2,2,2,2,3,3,4,4,5,5,5,5,5,6,6,6,6,10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 15, 15, 20, 20, 30, 30 ] We see that there is a unique conjugacy class in W (H ) of order 12. The class has 1200 elements. > C:=Conjugates(W,CC[28][3]); > #C; We set T := s s and look for an element T of order 12 such that s s T has charac- 1 1 3 ∞ 1 3 ∞ 4 3 2 2 teristic polynomial T − T − T + 1 = (T − 1) (T + T + 1). There are a fair number of such elements; we select for example: > Tinf; [ 0111] [a+1 a+1 1 0] [ -a-a-1 -1 0] [-a-1-a-1 -a -a] We can now construct the whole triple and compute the order of the group they generate. > T1:=s1*s3; > T0:=Tinfˆ(-1)*T1ˆ(-1); > T0; [ a100] [-a-1 -a-1 -1 -1] [2*a+12*a+2 a+2 a+1] [-a-1 -a-1 -a-1 -a] > T1; [-1000] [ a110] [0 0-1 0] [ 0011] > CharacteristicPolynomial(T0); xˆ4-xˆ3-x+1 > CharacteristicPolynomial(T1); xˆ4 - 2*xˆ2 + 1 > CharacteristicPolynomial(Tinf); xˆ4-xˆ2+1 > G:=sub<W|[T0,T1,Tinf]>; > #G; > IdentifyGroup(G); <144, 127> 38 Page 24 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 The monodromy group is then isomorphic to SL (F ) S acting via its four-dimensional 2 3 3 faithful irreducible representation with rational character. Note that this representation has Schur index 2 matching our obstruction calculation. Its smallest permutation repre- sentation is of degree 48. 10 Hurwitz example Some examples of G-II 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]. Consider the modular function u (τ):= η(7τ)/η(τ), where η(τ) is Dedekind’s eta- function. It is known that u is a Hauptmodul for X (7). As a function of t := 1728/j, where j is the standard elliptic j-invariant, u satisﬁes the algebraic equation 8 4 4 8 4 3 4 t(49u + 13u + 1)(7 u + 245u + 1) − 1728u = 0. 7 7 7 7 7 −1 It also satisﬁes a fourth-order diﬀerential equation of type G-II in terms of x = t 4 3 2 d u d u d u 7 7 7 2 2 2 x (x − 1) + x(x − 1)(7x − 4) + (573/56x − 5899/504x + 20/9) 4 3 2 dx dx dx du + (12297/5488x − 39779/24696) − 57/87808u = 0 (12) dx with exponents x exponents 0 0, 1/3, 2/3, 1 1 0, 1/2, 1, 3/2 ∞ −1/28, 3/28, 1/4, 19/28 We see that this example is a conjugate of that in row #8 of Table 1. Furthermore, consider the following modular functions for (7) 23/84 n −1 x(τ) = q (1 − q ) , n≡±4(mod7) 11/84 n −1 y(τ) = q (1 − q ) , n≡±2(mod7) −13/84 n −1 z(τ) = q (1 − q ) . n≡±1(mod7) 2 2 2 A full basis of solutions to (12) is then xyz, x y, y z, z x. Note that u = xyz. 11 A family of genus two curves In this section, we analyze in depth the G-II 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. Consider the G-II rigid local system G with parameters (α , α ) = (1/6, 5/6), (β , β ) = (1/6, 5/6), (γ , γ , γ , γ ) = (1/5, 2/5, 3/5, 4/5) 1 2 1 2 1 2 3 4 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 25 of 34 38 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 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 10 1 0 1010 −10 −1 −1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 111 1 0101 00 0 −1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T = ⎜ ⎟ ,T = ⎜ ⎟ ,T = ⎜ ⎟ . 0 1 ∞ ⎝ −100 −1 ⎠ ⎝ −1 000 ⎠ ⎝ 1000 ⎠ 000 1 0 −100 −11 0 0 Computing the invariant Hermitian form, we ﬁnd that these matrices are symplectic. Let :=T ,T ,T ⊆ Sp (Z) be the monodromy group. 0 1 ∞ We will show that G arises from H 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 equals the monodromy of a ﬁnite monodromy G-II modulo 2 (denoted G 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 (Z). ⎛ ⎞ ⎛ ⎞ 1000 00 −10 ⎜ ⎟ ⎜ ⎟ 1 −10 0 00 0 −1 ⎜ ⎟ ⎜ ⎟ K := ⎜ ⎟ ,L := ⎜ ⎟ . ⎝ 0011 ⎠ ⎝ 10 1 0 ⎠ 000 −1 01 0 0 In terms of these generators, we have −2 3 −1 −5 4 T = (KL ) L ,T = (KL ) . 1 ∞ We can easily verify using MAGMA that the monodromy group :=T ,T ,T ≤ 0 1 ∞ Sp (Z) is the unique subgroup of index two, namely the commutator subgroup of Sp (Z). 4 4 Here are the calculations > G<K,L>:=Group<K,L | Kˆ2=1, Lˆ12=1, K*Lˆ7*K*Lˆ5*K*L = L*K*Lˆ5*K*Lˆ7*K, Lˆ2*K*Lˆ4*K*Lˆ5*K*Lˆ7*K = K*Lˆ5*K*Lˆ7*K*Lˆ2*K*Lˆ4, Lˆ3*K*Lˆ3*K*Lˆ5*K*Lˆ7*K = K*Lˆ5*K*Lˆ7*K*Lˆ3*K*Lˆ3, (Lˆ2*K*Lˆ5*K*Lˆ7*K)ˆ2 = (K*Lˆ5*K*Lˆ7*K*Lˆ2)ˆ2, L*(Lˆ6*K*Lˆ5*K*Lˆ7*K)ˆ2 = (Lˆ6*K*Lˆ5*K*Lˆ7*K)ˆ2*L, (K*Lˆ5)ˆ5 = (Lˆ6*K*Lˆ5*K*Lˆ7*K)ˆ2>; > H<T1,Tinf> := sub<G | (K*Lˆ(-2))ˆ3*Lˆ(-1), (K*Lˆ(-5))ˆ4>; > Index(G,H); > H eq DerivedSubgroup(G); true The quotient of Sp (Z) by its level two congruence subgroup is isomorphic to Sp (F ), 4 4 which is known to be isomorphic to S .Wesee that maps surjectively to A under the 6 6 projection map f . > U:=SymmetricGroup(6); > homs := Homomorphisms(G, U : Limit := 1); > homs; Homomorphism of GrpFP: G into GrpPerm: U, Degree 6, Order 2ˆ4 * 3ˆ2 * 5 induced by a |--> (1, 2)(3, 4)(5, 6) b |--> (1, 2, 3)(4, 5) 38 Page 26 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 > f:=homs[1]; > f(T1); (1, 3, 2)(4, 5, 6) > f(Tinf); (1, 4, 6, 5, 3) > f(Tinfˆ(-1)*T1ˆ(-1)); (2, 3, 4) Consider now the G-II rigid local system G with parameters (α , α ) = (1/3, 2/3), (β , β ) = (1/3, 2/3), (γ , γ , γ , γ ) = (1/5, 2/5, 3/5, 4/5). 1 2 1 2 1 2 3 4 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 monodromy group is isomorphic to SL (F ), a 2 9 central extension of A by C . This group has two irreducible representations of degree 6 2 four with Schur index two. Here is the calculation with MAGMA. function goursat(q) K<a>:=GF(q); R<x>:=PolynomialRing(K); w:=RootsInSplittingField(xˆ2+x+1)[1][1]; T0:=[w + 1, 0, -1, 0, w, 1, -1, -1, w, 0, 0, (-w + 1)/w, 0, 0, 0, 1/w]; T1:= [-1, 0, 0, 1/w, -w - 1, -1, 1, 1/w, -w, -1, 0, (w +1)/w,-w, 0, 0, 0]; Tinf:=[w, 1/w, -1/w, (-wˆ2 - 1)/wˆ2, wˆ2 + 2*w + 1, 1/w, (-w - 1)/w, (-wˆ3 - wˆ2 - 1)/wˆ2, wˆ2 + w, (w +1)/w,(-w - 1)/w, (wˆ3 + wˆ2 +1)/ -wˆ2, wˆ2, 0, 0, -w]; G:=MatrixGroup<4,K|T0,T1,Tinf>; return G; end function; > G:=goursat(101ˆ2); > #G; > z:=IsIsomorphic(G,SL(2,9)); >z; true > Z:=Center(G); > #Z; > Z:=Center(G); > G/Z; Permutation group acting on a set of cardinality 6 Order = 360 = 2ˆ3 * 3ˆ2 * 5 (1, 2, 4) (1, 3, 2)(4, 5, 6) (2, 3, 4, 6, 5) Note that the parameters for G and G are equal up to fractions with denominator 2. This means that their respective local monodromies are the same modulo 2. It is clear, Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 27 of 34 38 for example, that the monodromy of G is isomorphic to A PSL (F ) modulo 2 as the 1 6 2 9 center acts by ±1. The three even permutations σ := f (T ) = (2, 3, 4), σ := f (T ) = (1, 3, 2)(4, 5, 6), σ := f (T ) = (1, 4, 6, 5, 3) 0 0 1 1 ∞ ∞ 3 2 we computed above generate A and correspond to a Belyi map with cycle type 31 , 3 , 51. D. Roberts showed us how this map is given by the following polynomial 3 3 2 P(x, t):= x (x + 3x − 5) − t(3x − 1). Indeed we have 3 3 2 P(x, 0) = x (x + 3x − 5), 2 3 P(x, 1) = (x + x − 1) , P(x, ∞) = 3x − 1. We consider the family of genus two curves deﬁned by the hyperelliptic equation C : y = 4P(x, t). Its Igusa invariants are 2 2 J = 2 · 3 · 5 · (4t + 1), 4 2 J = 2 · 3 · 5 · (4t + 1) , 2 4 3 2 J = 2 · 5 · (736t + 2928t − 564t + 25), 6 3 2 J = 3 · 5 · (4t + 1)(1856t − 8112t + 3156t − 25), 8 6 5 4 2 J = 2 · 3 · 5 · (t − 1) t . By construction, the Galois representation on the two torsion of the Jacobian of C 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 (C , Q) corresponds to G. We check that this indeed the case by computing the linear diﬀerential equation satisﬁed by periods of C . 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 ω. In this way, we ﬁnd that ω is annihilated modulo exact diﬀerentials by the diﬀerential operator 2 4 4 3 3 2 2 2 t (t − 1) D + t(t − 1) (9t − 5)D + (t − 1) (3456t − 3281t + 715)D 1 1 2 2 + (t − 1) (3888t − 2473)D + (186624t − 378373t + 169874). 450 810000 It is easy to verify that the diﬀerential equation is of the expected type G-II with exponents (−1/6, 0, 1/6, 1), (−1/6, 1/6, 5/6, 7/6), (2/5, 3/5, 4/5, 6/5) at t = 0, 1, ∞, respectively. 38 Page 28 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Table 8 Inﬁnite families β, α , α γ/AA |Z()| Impr 1 2 11/2, 1/2,r −r/4, 1/4 − r/4, 1/2 − r/4, 3/4 − r/4 D n ∗ 4 1,n 21/2, 1/3, 2/3 r, −r/3, 1/3 − r/3, 2/3 − r/3 A gcd(n, 4) ∗ 4 2,n 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 ⊆ SL (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). In this table, r = m/n is an arbitrary rational number in the lowest terms and C,v (n) = 0, ⎪ 2 ⎨ 4 ⎨ C v (n) = 0, ≡ ≡ C C,v (n) = 1, 1,n 2,n 2 2 n/2 ⎩ ⎪ C C v (n) ≥ 1, ⎪ n 2 n/2 ⎩ C C,v (n) ≥ 2, 4 2 n/4 where v is the valuation at 2. 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 for some n.For all of these systems, there are inﬁnite families of examples all lying in a single geodesic in irr the positive components T . 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 ∈ Q the function r r f satisﬁes L f = 0,where L is a diﬀerential operator of order m, whose coeﬃcients r r depend polynomially on r. Proof Let P(t, y) = 0 be the deﬁning equation for f ,andlet y (t), ... ,y (t) be its solutions. 1 m Denote by W (f , ... ,f ) the Wronskian determinant and let us write 1 n r r W (y, y , ... ,y ) 1 (m) (m−1) L [y] = = y + A y + ··· + A . r m−1 0 W (y , ... ,y ) Deﬁne polynomial diﬀerential operators D by (n) f f i i = D . f f i i Then D (f ) = 1, D (f ) = f , D (f ) = f + f , and in general they are deﬁned by the recur- 0 1 2 sion D (f ) = (D (f )) + D (f )f . Using these operators we can write the Wronskian in n+1 n n terms of logarithmic derivatives as W (f , ... ,f ) = f ... f det(D (f /f )) . Expanding the 1 n 1 n i j i,j determinants in the deﬁnition of L shows that A can be expressed as rational functions r k in r whose coeﬃcients are symmetric expressions in y , ... ,y and their derivatives and 1 m thus are rational functions of t. r r r r For generic r (more precisely, whenever W (y , ... ,y ) = 0) the functions y , ... ,y 1 m 1 m form the full space of solutions of L , and thus, the singularities of L are contained in the r r Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 29 of 34 38 Fig. 1 r r set of singularities of y , ... ,y together with the set of points t where one of y becomes 0. In terms of the deﬁning equation P(t, y) = p (t)y + ··· + p (t), these are exactly the m 0 values of t where p (t)p (t)orthe discriminant of P vanishes. 0 m For instance, consider u (τ):= (η(2τ)/η(τ)) .Itisknown that u is a Hauptmodul for 2 2 (2) and satisﬁes the algebraic equation A (u, t):= t(1 + 256u) − 1728u = 0,t := 1728/j, where j is the standard elliptic j-invariant. For any r ∈ Q,its r-th power is annihilated by the third-order diﬀerential operator 3 2 2 d d 20 (3r + 2)(r − 1) d r (r − 1) 3 2 x (x − 1) + x (4x − 5/2) + x x + + , 3 2 dx dx 9 4 dx 4 when expressed in x = t(τ) = 1728/j(τ). The local exponents are 0:(r, −r/2, (1 − r)/2), 1:(0, 1/2, 1), ∞ :(0, 1/3, 2/3). Hence these are hypergeometric equations. 12.2 Simpson even and odd families Given an odd positive integer N > 1 consider the hypergeometric series N −1 N +1 2N 2N f (t):= F | t . N 2 1 It is an algebraic function of t satisfying P (f (t),t) = 0 for a polynomial P (u, t) ∈ Q[u, t]. N N N This polynomial P can be given explicitly; the information we need is the shape of its 2 r s Newton polygon , convex hull of the (r, s) ∈ Z for which the monomial u t in P has a N N nonzero coeﬃcient. The polygon is in fact the triangle of vertices (0, 0), (1, 0), (N, (N − 1)/2). If we orient the boundary of the triangle counterclockwise starting at the origin the three sides have slopes 0, 1/2, −κ , where κ := (N − 1)/2N, respectively. N N For example, for N = 5weﬁnd 5 2 3 P (u, t) = 16u t − 500u t + 3125u − 3125 and its Newton polygon is with sides of slope 0, 1/2, −2/5 (see Fig. 1). At a zero or pole of f (t)wehave t = 0,1or t =∞. Hence by Proposition 4, f (t) N N 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. 38 Page 30 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 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 1 e − 1 [κ]:= 0 − κr, − κr, ... , − κr, d d where d is the denominator of κ and e is the horizontal width of δ. The exponents at t = 0 for f are [κ ], [κ ], ..., where κ , κ , ... runs over the slopes of 1 2 1 2 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 p(u, t + 1). In the case of f the bottom slopes are 0, 1/2, the only top slope is −κ and we obtain N N the following t exponents 0 0, −r/2, 1/2 − r/2, ... , (N − 2)/2 − r/2 1 0, 1/2, 1, ... , (N − 1)/2 ∞ κ r, 1/N + κ r, ... , (N − 1)/N + κ r N N N For example, when N = 5 these exponents are t exponents 0 0, −r/2, 1/2 − r/2, 1 − r/2, 3/2 − r/2 1 0, 1/2, 1, 3/2, 2 ∞ 2/5r, 1/5 + 2/5r, 3/5 + 2/5r, 3/5 + 2/5r, 4/5 + 2/5r If r ∈ Q is not an integer then the multiplicity of these exponents is (m, m, 1), (m + 1,m), (1, ... , 1), where m := (N − 1)/2. These are precisely the multiplicities of Simpson’s odd rank case family of rigid local systems. Hence we have obtained a geodesic completely irr contained in the positive components T of this system’s parameter space. A completely analogous discussion holds for N even with the same deﬁnition of f .Here it is more convenient to consider the algebraic equation for f , which is the hypergeometric function N −1 N +1 N N f (t) = F |t . N 2 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 multiplic- ities of Simpson’s even rank case family of rigid local systems [24]. Again we have obtained irr a geodesic completely contained in the positive components T of the parameter space. For example, for N = 4 we get a geodesic for Goursat G-II up to a twist. t exponents 0 0, −r, 1/2 − r, 1 − r 1 0, 1/2, 1, 3/2 ∞ 3/4r, 1/4 + 3/4r, 1/2 + 3/4r, 3/4 + 3/4r 12.3 Simpson extra case of rank 6 The Hauptmodul u := (η(5τ)/η(τ)) satisﬁes the equation 2 3 A (u, t) = t(3125u + 250u + 1) − 1728u = 0, 5 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 31 of 34 38 whose Newton polygon is a triangle with vertices (0, 1), (4, 1), (0, 1) and slopes −1, 1/5, 0. Fractional powers of u (τ) = (η(5τ)/η(τ)) give the rank 6 extra case rigid local system of Simpson. Explicitly, in terms of x = 1/t(τ), u satisﬁes the diﬀerential equation 6 5 d d 4 2 3 x (x − 1) + x (x − 1)(17x − 10) 6 5 dx dx 2 2 4 −3r + 2r + 432 108r − 72r − 18377 220 d 2 2 + x x + x + 5 180 9 dx 3 2 8r − 118r + 74r + 3720 + x x 3 2 3 576r − 5796r + 3528r + 209915 40 d − x + 1800 3 dx 4 3 2 −45r + 860r − 5145r + 2950r + 45024 + x 3 2 2 576r − 2676r + 1448r + 26135 40 d − x + 900 81 dx 5 4 3 2 24r − 415r + 2790r − 7925r + 3846r + 15120 + x 3 2 64r − 164r + 72r + 315 d 900 dx r (r − 1)(r − 2)(r − 3)(r − 4) − . The exponents of this equation for generic r ∈ Q are t exponents 0 r, −r/5, 1/5 − r/5, 2/5 − r/5, 3/5 − r/5, 4/5 − r/5 1 0, 1/2, 1, 3/2, 2, 3 ∞ 0, 1/3, 2/3, 1, 4/3, 5/3 with multiplicities (4, 2), (2, 2, 2), (1, ... , 1). This is then a geodesic in the positive compo- irr nents T of Simpson’s extra case. A special case reduces to a hypergeometric series 1 1 5 2 6 6 = F |t . 3 2 4 6 5 5 For r = 1, 3, the equation reduces to a hypergeometric equation of order 3, and for r = 2, 4, 8, 14 it reduces to an equation of order 5 with rigid monodromy of Simpson’s odd type. 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. Consider the algebraic equation u(1 − u) − t = 0. 5 38 Page 32 of 34 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 We can solve for u as a function of t by inversion. Let 5 5 5 u(t) 13 51 1771 4095 5 2 3 4 5 6 f (t) = = 1 + t + t + t + t + t + O(t ),t := t. 1 1 1 1 1 1 1 4 4 4 t 8 16 256 256 4 We ﬁnd that 4 6 7 8 5 5 5 5 f (t) = F |t . 4 3 3 5 7 2 4 4 We compute the local exponents of f for r ∈ Q and ﬁnd t exponents 0 0, −r, 1/4 − r, 1/2 − r, 3/4 − r 1 1/2, 0, 1, 2, 3 ∞ 4/5r, 1/5 + 4/5r, 2/5 + 4/5r, 3/5 + 4/5r, 4/5 + 4/5r We obtain the following identity between hypergeometric functions 4 6 7 8 4/5r 1/5 + 4/5r 2/5 + 4/5r 3/5 + 4/5r 4/5 + 4/5r 5 5 5 5 F |t = F |t . 4 3 5 4 3 5 7 2 1 + r 3/4 + r 1/2 + r 1/4 + r 1 2 4 4 12.5 Goursat II There is another geodesic in the case G-II apart from that in Sect. 12.2 for n = 4. Consider the modular unit u (τ):= (η(3τ)/η(τ)) . It is a classical fact that u is a Hauptmodul 3 3 6 −1 for the modular curve X (3) and satisﬁes the algebraic equation A (3 u , 1728j ) = 0, 0 3 3 where A (u, t):= t(u + 27)(u + 3) − 1728u. The Newton polygon of A is a triangle with vertices (0, 1), (4, 1), (0, 1) and slopes −1, 1/3, 0. The fourth-order diﬀerential equation satisﬁed by u is 4 3 d d 2 2 x (x − 1) + x(x − 1)(7x − 4) 4 3 dx dx 2 2 2 −6r + 3r + 92 24r − 12r − 421 20 d + x + x + 9 36 9 dx 3 2 3 2 8r − 33r + 13r + 60 64r − 192r + 68r + 345 d + x − 27 216 dx r (r − 1)(r − 2) − , (13) when expressed in x = 1/t(τ) = j(τ)/1728. The exponents of this equation for generic r ∈ Q are t exponents 0 0, 1/3, 2/3, 1 1 0, 1/2, 1, 3/2 ∞ r, −r/3, 1/3 − r/3, 2/3 − r/3 We have 7 1 u 1728 12 4 = F |t ,t := . 2 1 t 1 j Here is a detailed description of this geodesic. The four exponents at ∞ are γ = 3r, γ =−r, γ =−r + 1/3, γ =−r + 2/3. 1 2 3 4 Radchenko and Rodriguez Villegas Res Math Sci (2018) 5:38 Page 33 of 34 38 0 1 Fig. 2 G-II geodesic γ 0 1 Fig. 3 G-II geodesic δ We plot these modulo Z as a function of r (see Fig. 2). The ﬁrst condition (n = 2inthe irr notation of (3)) for these parameters for generic r to be in the positive components T is that there are two in each of the indicated horizontal strips. This is visible in the plot. The second condition for positive deﬁniteness involves the parameters δ = 2r, δ = 2r + 2/3, δ = 2r + 1/3, 1 2 3 δ =−2r + 2/3, δ =−2r + 1/3, δ =−2r. 4 5 6 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 Fig. 3 where this condition is visible. Author details 1 2 Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany, The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy. 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. 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Research in the Mathematical Sciences – Springer Journals
Published: Sep 5, 2018
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