Towards a nonlinear Schwarzs list

T o wards a nonlinear Schw arz’ s list Philip Boalch 1 Intr oduction The main theme of this talk is “icosahedral” solutions of (ordinary) dif ferential equations, a topic that seems suitable for a 60th birthday conference. W e will ho wev er try to go beyond the icosahedron, to see what comes next, and consider various symmetry groups each of which could be thought of as the next in a sequence, follo wing the icosahedral group. T o fix ideas let us giv e a classical example. Recall the icosahedral rotation group of order 60: A 5 ∼ = PSL 2 ( F 5 ) ∼ = ∆ 235 ∼ = h a , b , c   a 2 = b 3 = c 5 = abc = 1 i . This is described via three generators a , b , c whose product is the identity , and so it is natural to look for ordinary dif ferential equations on the 3 punctured sphere P 1 ( C ) \ { 0 , 1 , ∞ } with monodromy group A 5 . No w A 5 is a three-dimensional rotation group so naturally liv es in SO 3 ( R ) which is a subgroup of SO 3 ( C ) which is isomorphic to PSL 2 ( C ) . Thus we are led to search for connections ∇ = d −  A 1 z + A 2 z − 1  d z , A i ∈ sl 2 ( C ) (1) on rank two holomorphic vector bundles ov er the three-punctured sphere with pr ojective monodromy group equal to A 5 . Such connections are essentially the same as Gauss hyper geometric equations, and H. Schwarz [49] classified all such equations having finite monodromy groups in 1873. The list he produced has 15 rows, one for the family of dihedral groups, two rows for each of the tetrahedral and octahedral groups and 10 ro ws for the icosahedral group. See T able 1. A ke y point here is that the Gauss hyper geometric equation is rigid so the full monodromy representation (of the fundamental group of the 3-punctured sphere into PSL 2 ( C ) ) is de- termined by the conjugacy classes of the monodromy around each of the punctures. Thus in Schwarz’ s list it is suf ficient to list these local monodromy conjugac y classes in order to specify the possible monodromy representations (and from this it is easy to find a hyperge- ometric equation with gi ven monodromy). T o ease recognition, to the left of the table we 1 T able 1: Schwarz’ s list [49] hav e listed the triples of conjugacy classes which occur , labelling the four nontrivial con- jugacy classes of A 5 by a , b , c , d , representing rotations by 1 2 , 1 3 , 1 5 , 2 5 of a turn, respectively . (In the octahedral case one may also hav e rotations by a quarter of a turn, which we label by g .) 1.1 Naiv e generalisations Our basic aim is to discuss three naiv e generalisations of Schwarz’ s list, as follo ws. The first two arise simply by looking for nonrigid connections that are natural generalisations of the hyper geometric connections considered above, obtained by adding an extra singularity— the two cases are generalisations of two ways one may view the hypergeometric equation as a connection. First of all we can simply add another pole at some point t : ( A ) ∇ = d −  A 1 z + A 2 z − t + A 3 z − 1  d z , A i ∈ sl 2 ( C ) and keep the coef ficients in sl 2 ( C ) . 2 Secondly we recall that the connection one obtains immediately upon choosing a cyclic vector for the hypergeometric equation is as in (1) but with A 1 , A 2 both rank one matrices (in gl 2 ( C ) ). Then the monodromy group will be a complex reflection group (generated by tw o two-dimensional complex reflections 1 ) and the natural generalisation is then to consider connections of the form: ( B ) ∇ = d −  B 1 z + B 2 z − t + B 3 z − 1  d z , B i ∈ gl 3 ( C ) with each B i having rank one, so the monodromy group will be generated by three three- dimensional complex reflections. This is a very natural condition as we will see. Questions A, B: Find the analogue of Schwarz’ s list for connections (A) or (B) . These questions can no w be answered and lead to two “nonrigid Schwarz’ s lists”, i.e. to classifications of possible monodromy representations with finite image (up to equiv alence) and the construction of connections realizing such representations. W e should emphasise that the main focus has been the construction of such connections with giv en monodromy representation for any v alue of t (which is a tricky business in this nonrigid case), rather than just the classification (which is reasonably straightforward). Example (of type (B) ). The full symmetry group of the icosahedron is the icosahedral reflection group of order 120: H = H 3 ∼ = h r 1 , r 2 , r 3   r 2 i = 1 , ( r 1 r 2 ) 2 = ( r 2 r 3 ) 3 = ( r 3 r 1 ) 5 = 1 i ⊂ O 3 ( R ) ⊂ GL 3 ( C ) . This is generated by three reflections (whose product is not the identity) and so it is natural to look for connections on rank three bundles over a four-punctured sphere with mon- odromy H (generated by 3 reflections about 3 of the punctures—i.e. connections of the form (B) with each of the three residues B i having trace 1 2 so the corresponding reflections are of order two). There turn out to be three inequiv alent triples of generating reflections of H , two of which are related by an outer automorphism. The problem is to write down connections of the desired form for any value t of the final pole position. One triple of generating reflections is intimately related to K. Saito’ s flat structure for H (or icosahedral Frobenius manifold) [48] and appears in Dubrovin’ s article [19] Appendix E. The other two triples were dealt with around 1997 by Dubrovin and Mazzocco [21]; one is similar to the first case (since related to it by an outer automorphism) b ut the final triple turned out to be much trickier , and writing out the family of connections in this case in volved a spe- cific elliptic curve which took about ten pages of 40 digits integers to write down (see the preprint version of [21] on the mathematics arxiv). W e will ev entually see below that this elliptic solution is in fact equiv alent to a solution with a simple parametrisation, agreeing with Hitchin’ s philosophy that “nice problems should hav e nice solutions”. 1 i.e. arbitrary automorphisms of the form “one plus rank one”, not necessarily of order two or orthogonal. 3 Remark Before moving on to the third generalisation let us add some other historical comments. The “non-naiv e” generalisations of the Gauss hypergeometric equation are the equations satisfied by the n F n − 1 hyper geometric functions (the Gauss case being that of n = 2). The corresponding Schwarz’ s list appears in the 1989 article [4] of Beukers and Heckman. In terms of connections this amounts to considering connections (1) on rank n vector bundles, still with three singularities on P 1 , but with A 1 of rank n − 1 and A 2 of rank 1; these connections ar e still rigid . Some work in the nonrigid case has been done (besides that we will recall belo w) by considering generalisations of the hypergeometric equation as an equation (rather than as a connection); for example the algebraic solutions of the Lam ´ e equation were studied in [5] by Beukers and van der W aall (Lam ´ e equations are basically the second order Fuchsian equations with four singular points on P 1 such that three of the local monodromies are of order two). In general connections of type (A) with such monodromy representations will not come from a Lam ´ e equation (since upon choosing a cyclic vector the corresponding equations will in general ha ve additional apparent singularities; this can also be seen by counting dimensions). Indeed it turns out ([5]) that Lam ´ e equations only have finite monodromy for special configurations of the four poles. 1.2 Nonlinear analogue—the Painle v ´ e VI equation One reason hypergeometric equations are interesting is that they provide the simplest ex- plicit examples of Gauss–Manin connections . Indeed this is one reason Gauss was inter- ested in them: he observed that the periods of a family of elliptic curves satisfy a (Gauss) hyper geometric equation. (The modern interpretation of this is as the explicit form of the natural flat connection on the vector bundle of first cohomologies over the base of the family of elliptic curves, written with respect to the basis gi ven by the holomorphic one forms—and their deri vati ves—on the fibres.) Nowadays there is still much interest in such linear dif ferential equations “coming from geometry”. Thus the nonlinear analogue of the Gauss hypergeometric equation should be the explicit form of the simplest nonabelian Gauss–Manin connection (i.e the explicit form of the natu- ral connection on the bundle of first nonabelian cohomologies of some family of v arieties). The simplest interesting case corresponds to taking the univ ersal family of four punctured spheres and taking cohomology with coef ficients in SL 2 ( C ) (one needs a non-tri vial f amily of v arieties with nonabelian fundamental groups). This leads to the Painle v ´ e VI equation (P VI ), which is a second order nonlinear differential equation whose solutions, like those of the hypergeometric equation, branch only at 0 , 1 , ∞ ∈ P 1 . In particular we may study the (nonlinear) monodromy of solutions of P VI , by examining ho w solutions v ary upon analytic continuation along paths in the three-punctured sphere. Thus, since Schwarz lists fundamental solutions of hyper geometric equations having finite monodromy , our main question is to construct the analogue of Schwarz’ s list for P VI : Question C: What are the solutions of Painle v ´ e VI having finite monodromy? This question is still open; there is as yet no full classification—the main ef fort (at least of the present author) has been tow ards finding and constructing interesting solutions. So 4 far all kno wn finite branching solutions are actually algebraic 2 . Currently we are at the reasonably happy state of affairs that all such solutions kno wn to exist hav e actually been constructed. In what follo ws I will explain v arious aspects of the problem, and in particular sho w ho w the nonrigid lists (A) or (B) map to the list of (C) . Some ke y points, demonstrat- ing the richness and v ariety of solutions, are: • There are algebraic solutions of P VI not related to finite subgroups of the coef ficient group SL 2 ( C ) , • There are ‘generic’ solutions of P VI with finite monodromy; i.e. not lying on any of the reflection hyperplanes of the af fine F 4 W eyl group of symmetries of P VI , • There are entries on the list of (C) which do not come from either (A) or (B) . In particular we will see P VI solutions related to the groups A 6 , PSL 2 ( F 7 ) and ∆ 237 . 2 What is Painle v ´ e VI? There are various viewpoints, and simply gi ving the explicit equation is perhaps the least helpful introduction to it. In brief, Painle v ´ e VI is: • the explicit form of the simplest nonabelian Gauss–Manin connection, • the equation controlling the “isomonodromic deformations” of certain logarithmic con- nections/Fuchsian systems on P 1 , • the most general second order ODE with the so-called ‘Painle v ´ e property’, • a certain dimensional reduction of the anti-self-dual Y ang–Mills equations (see e.g. [41]), • an equation related to certain elliptic integrals with moving endpoints (cf. R. Fuchs [23] and Manin [40]), • The second order ODE for a complex function y ( t ) d 2 y d t 2 = 1 2  1 y + 1 y − 1 + 1 y − t   d y d t  2 −  1 t + 1 t − 1 + 1 y − t  d y d t + y ( y − 1 )( y − t ) t 2 ( t − 1 ) 2  α + β t y 2 + γ ( t − 1 ) ( y − 1 ) 2 + δ t ( t − 1 ) ( y − t ) 2  where α , β , γ , δ ∈ C are constants. The Painle v ´ e property means that any local solution y ( t ) defined in a disc in the three- punctured sphere P 1 \ { 0 , 1 , ∞ } extends to a meromorphic function on the univ ersal cover 2 Apparently Iwasaki [28] has recently sho wn that all finite branching solutions are algebraic. 5 of P 1 \ { 0 , 1 , ∞ } . It is this property that enables us to speak of the monodromy of P VI solutions. Concerning solutions there is a basic trichotomy (see W atanabe [53]): A solution of P VI is either      a ‘ne w’ transcendental function, or a solution of a 1st order Riccati equation, or an algebraic function. In particular if one is interested in constructing ne w explicit solutions of P VI then, since the Riccati solutions are all well understood, the algebraic solutions are the first place to look. The standard approach to P VI is as isomonodromic deformations of rank two logarithmic connections with four poles on P 1 , as the poles mov e (generic such connections are of the form (A) , and then t parametrises the possible pole configurations). In particular one can see the four constants in P VI directly in terms of the eigen values of the residues of the connection: if we set θ i to be the difference of the eigen v alues (in some order) of the residue A i ( i = 1 , 2 , 3 , 4, where A 4 = − ∑ 3 1 A i is the residue at infinity) then the relation to the constants is α = ( θ 4 − 1 ) 2 / 2 , β = − θ 2 1 / 2 , γ = θ 2 3 / 2 , δ = ( 1 − θ 2 2 ) / 2 . Before going into more detail let us also mention one further property of P VI : it admits a group of symmetries isomorphic to the affine W eyl group of type F 4 (see [45] or the exposition in [10]). Indeed treating θ = ( θ 1 , . . . , θ 4 ) ∈ C 4 as the set of parameters for P VI is useful since the affine F 4 W eyl group of symmetries acts in the standard way on this C 4 . (W e will see belo w that these four parameters may also be interpreted as coordinates on the moduli space of cubic surfaces.) 2.1 Conceptual appr oach to P VI Consider the uni versal family of smooth four -punctured rational curv es with labelled punc- tures. Write B : = M 0 , 4 ∼ = P 1 \ { 0 , 1 , ∞ } for the base, F for the standard fibre and C for the total space: C ← − − − F ∼ = P 1 \ 4 points   y B No w replace each fibre F by H 1 ( F , G ) where G = SL 2 ( C ) . Here we will use two vie w- points/realisations of this nonabelian cohomology set H 1 : 1) (Betti): Moduli of fundamental group representations: 6 H 1 ( F , G ) ∼ = Hom ( π 1 ( F ) , G ) / G 2) (De Rham): Moduli of connections on holomorphic vector bundles o ver F These two vie wpoints are related by the Riemann–Hilbert correspondence (the nonabelian De Rham functor), taking connections to their monodromy representations. The point is that algebraically these realisations of H 1 are very different and the Riemann–Hilbert map is transcendental (things written in algebraic coordinates on one side will look a lot more complicated from the other side). Thus we get two nonlinear fibrations over the base B , with fibres the De Rham or Betti realisations of H 1 ( F , G ) respecti vely: M De Rham Riemann–Hilbert − − − − − − − − − → M Betti   y   y B B As in the case with abelian coefficients one still gets a natural connection on these coho- mology b undles. The surprising fact is that it is algebraic on both sides (approximating the De Rham side in terms of logarithmic connections to giv e it an algebraic structure [44]). Thus when written explicitly we will get nonlinear algebr aic dif ferential equations “com- ing from geometry”. (See Simpson [51] Section 8 for more on these connections in the case of families of projecti ve v arieties.) The two standard descriptions of the abelian Gauss–Manin connection generalise to de- scriptions of this nonlinear connection. In the Betti picture we may identify two nearby fibres of M Betti simply by keeping the monodromy representations (points of the fibres) constant: moving around in B amounts to deforming the configuration of four points in P 1 and it is easy to see how to identify the fundamental groups of the four punctured spheres as the punctures are deformed—use the same generating loops. This ‘isomonodromic’ de- scription, preserving the monodromy representation, is the nonabelian analogue of keeping the periods of one-forms constant. On the De Rham side the nonlinear connection can be described in terms of extending a connection on a v ector bundle ov er a fibre F , to a flat connection on a vector bundle over a family of fibres and then restricting to another fibre, much as the abelian case is described in terms of closed one-forms (linear connections replacing one-forms and flatness replacing the notion of closedness). Each of these descriptions has a use: the De Rham viewpoint lends itself to giving an explicit form of the nonlinear connection (essentially amounting to the condition for the flatness of the connection over the family of fibres). The Betti viewpoint is more global 7 and allo ws us to study the monodromy of the nonlinear connection, as an explicit action on fibres of M Betti . 2.2 Explicit nonlinear equations The De Rham bundle M De Rham is well approximated by the space of logarithmic connec- tions with four poles on the trivial rank two holomorphic bundle (with trivial determinant) ov er P 1 . Call the space of such connections M ∗ and observe it parametrises connections of the form (A) , and that these are determined by the v alue of t ∈ B and the residues: M ∗ ∼ = B ×  ( A 1 , . . . , A 4 )   A i ∈ g , ∑ A i = 0  / G . Here G = SL 2 ( C ) does not act on B and acts by diagonal conjugation on the residues A i . In general this quotient will not be well behaved, b ut it has a natural Poisson structure and the generic symplectic leav es will be smooth complex symplectic surfaces. Clearly M ∗ is tri vial as a bundle ov er B (projecting onto the configuration of poles), but the nonabelian Gauss–Manin connection is dif ferent to this tri vial connection and was computed about 100 years ago by Schlesinger (essentially in the way stated above it seems). The nonlinear connection is gi ven by Schlesing er’ s equations , which in the case at hand are: d A 1 d t = [ A 2 , A 1 ] t , d A 3 d t = [ A 2 , A 3 ] t − 1 together with a third equation for d A 2 / d t easily deduced from the fact that A 4 remains constant. If the residues of the connection satisfy these equations then the corresponding monodromy representation remains constant as t varies. (They are easily deriv ed from the v anishing of the curvature of the ‘full’ connection d −  A 1 d z z + A 2 d z − d t z − t + A 3 d z z − 1  .) T o get from here to P VI one chooses specific functions x , y on M ∗ which restrict to coor- dinates on each generic symplectic leaf and writes down the connection in these (carefully chosen) coordinates. (See [8] pp.199-200 for a discussion of the formulae, which are from [31].) This leads to two coupled nonlinear first order equations, and eliminating x leads to the second order Painle v ´ e VI equation for y ( t ) . It was first written down in full generality by R. Fuchs [23] (whose father L. Fuchs was also the f ather ‘Fuchsian equations’). 2.3 Monodr omy of Painle v ´ e VI Since the Betti and De Rham realisations are isomorphic, we see the monodromy of solu- tions to P VI thus corresponds to the monodromy of the connection on M Betti . This amounts to an action of the fundamental group of the base B on a fibre, and this action can be described explicitly . 8 Let M t = Hom ( π 1 ( P 1 \ { 0 , t , 1 , ∞ } ) , G ) / G be the fibre of M Betti at some fixed point t ∈ B . The key point is that π 1 ( B ) ∼ = F 2 (the free nonabelian group on two generators) may be identified with the pure mapping class group of the four punctured sphere P 1 \ { 0 , t , 1 , ∞ } . As such it has a natural action on M t (by pushing forward loops generating the fundamental group), and this action is the desired monodromy action. Explicitly , upon choosing appropriate generating loops of π 1 ( P 1 \ { 0 , t , 1 , ∞ } ) we see M t may be described directly in terms of monodromy matrices: M t ∼ =  ( M 1 , . . . M 4 )   M i ∈ G , M 4 · · · M 1 = 1  / G which in turn is simply the quotient G 3 / G of three copies of G by diagonal conjugation by G = SL 2 ( C ) . In fact this quotient has been studied classically: the ring of G in vari- ant functions on G 3 has 7 generators and one relation, embedding the affine GIT quotient as a hypersurface in C 7 . The particular equation for this hypersurface appears on p.366 of the book [22] of Fricke and Klein. The Painle v ´ e VI parameters essentially specify the conjugacy classes of the four monodromies M i , and serve here to fibre the six-dimensional hypersurface G 3 / G into a four parameter family of surfaces. Looking at the explicit equa- tion sho ws they are affine cubic surfaces. In turn Iwasaki [29] has recently pointed out that this family of cubics may be quite simply related to Cayley’ s explicit family [14] and so contains the generic cubic surface. The desired action of the free group F 2 on the Betti spaces is giv en by the squares of the follo wing “Hurwitz” action: ω 1 ( M 1 , M 2 , M 3 ) = ( M 2 , M 2 M 1 M − 1 2 , M 3 ) ω 2 ( M 1 , M 2 , M 3 ) = ( M 1 , M 3 , M 3 M 2 M − 1 3 ) . More explicitly if we consider simple positiv e loops l 1 , l 2 in B based at 1 2 encircling 0 , 1 resp. then the monodromy of the connection on M Betti around l i is giv en by ω 2 i (with respect to certain generators of π 1 ( P 1 \ { 0 , 1 2 , 1 , ∞ } ). In turn it is possible to write this action directly as an action on the ring of in v ariant function on G 3 . 3 Algebraic solutions fr om finite subgroups of SL 2 ( C ) 3.1 What exactly is an algebraic solution? The obvious definition is simply an algebraic function y ( t ) which solves P VI for some v alue of the four parameters. Thus it will be specified by some polynomial equation F ( y , t ) = 0 . 9 and a four-tuple θ of parameters. In practice ho wev er such polynomials F can be quite unwieldy and are difficult to transform under the affine W eyl symmetry group, making it dif ficult to see if in fact two solutions are equi v alent. This leads to our preferred definition: Definition. An algebraic solution of P VI is a compact, possibly singular , algebraic curve Π together with two rational functions y , t : Π → P 1 Π y − − − → P 1 t   y P 1 such that • t is a Belyi map (i.e. its branch locus is a subset of { 0 , 1 , ∞ } ), and • y , when viewed as a function of t aw ay from the ramification points of t , solves P VI for some v alue of the four parameters. In principle it is straightforward to go between the two definitions, but in practice it is useful to look for a good model of Π (and the model gi ven by the closure of the zero locus of the polynomial F is usually a bad choice). 3.2 (A) 7→ (C) Suppose we hav e a linear connection (A) with finite monodromy . Its monodromy represen- tation will be specified by a triple ( M 1 , M 2 , M 3 ) ∈ G 3 generating a finite subgroup Γ ⊂ G (where G = SL 2 ( C ) as above). This linear connection specifies the initial v alue (and first deri vati ve) of a solution to P VI . This P VI solution will have finite monodromy , since we kno w the branching of P VI solutions corresponds to the F 2 action on conjugacy classes of triples in G 3 , and the orbit through ( M 1 , M 2 , M 3 ) will be finite, since the action is within triples of generators of Γ . Thus we see that finite F 2 orbits (in G 3 / G ) correspond to P VI solutions with a finite number of branches, and the points of such F 2 orbits correspond to the individual branches of the P VI solution. In particular the size of the orbit, the number of branches, is the degree of the map t : Π → P 1 . (Indeed the F 2 action on such a finite orbit itself gi ves the full permutation representation of the Belyi map t : Π → P 1 , and in particular , by the Riemann–Hurwitz formula, determines the genus of the ‘Painle v ´ e curve’ Π .) Said dif ferently it is useful to define a topolo gical algebr aic P VI solution (or henceforth for bre vity a topological solution ) to be a finite F 2 orbit in G 3 / G . (The classification of such orbits is still open and is the main step in classifying all finite branching P VI solutions.) In these terms the first paragraph above points out that one obtains “obvious” topological 10 solutions upon taking any triple of generators of an y finite subgroup of G . For example (omitting discussion of ho w they were actually constructed) here are some solutions corresponding to certain triples of generators of the binary tetrahedral and octa- hedral subgroups, due to Dubrovin [19] and Hitchin [27] (in different but equiv alent forms): T etrahedral solution of degree three y = ( s − 1 )( s + 2 ) s ( s + 1 ) , t = ( s − 1 ) 2 ( s + 2 ) ( s + 1 ) 2 ( s − 2 ) , θ = ( 2 , 1 , 1 , 2 ) / 3 , Octahedral solution of degree four y = ( s − 1 ) 2 s ( s − 2 ) , t = ( s + 1 )( s − 1 ) 3 s 3 ( s − 2 ) , θ = ( 1 , 1 , 1 , 1 ) / 4 . In both cases Π is a rational curve (with parameter s ). Although written in this compact form, one should bear in mind these formulae represent a whole (isomonodromic) family of connections (A) as t varies. An e xplicit elliptic solution appears in Hitchin [25] and may be written as: Elliptic dihedral solution θ = ( 1 , 1 , 1 , 1 ) / 2 y = ( 3 s − 1 )  s 2 − 4 s − 1   s 2 + u  ( s ( s + 2 ) − u ) ( 3 s 3 + 7 s 2 + s + 1 ) ( s 2 − u ) ( s ( s − 2 ) + u ) , t =  s 2 + u  2 ( s ( s + 2 ) − u ) ( s ( s − 2 ) − u ) ( s 2 − u ) 2 ( s ( s + 2 ) + u ) ( s ( s − 2 ) + u ) where the pair ( s , u ) li ves on the elliptic curve u 2 = s  s 2 + s − 1  . This solution has de gree 12 and corresponds to a triple of generators of the binary dihedral group of order 20. It turns out (see [9] Remark 16) that the icosahedral solutions of Dubrovin and Mazzocco [21] fit into this framew ork as well and correspond to (certain) triples of generators of the binary icosahedral group, although in the first instance they arose from the icosahedral re- flection group as described earlier . Note that [21] Remark 0.1 describes a relation between their solutions of P VI and a certain folding of Schwarz’ s list; this is dif ferent to the relation just mentioned—in particular problem (A) demands an extension of Schw arz’ s list. 4 Bey ond Platonic P VI solutions My starting point in this project was simply the observation that there should be more al- gebraic solutions to P VI than those coming from finite subgroups of SL 2 ( C ) . Dubro vin [19] had shown how to relate three dimensional real orthogonal reflection groups to a cer- tain one-parameter family of the full four dimensional family of P VI equations (namely the 11 family having parameters θ = ( 0 , 0 , 0 , ∗ ) ) and this was used in [21] to classify algebraic solutions ha ving parameters in this one-parameter f amily . (Some aspects of [21] were sub- sequently extended by Mazzocco in [42] to classify rational solutions—i.e. those with only one branch, cf. also [54].) The further observation w as that if one is able to get away from the orthogonality condition here then one will relate any P VI equation to a three dimen- sional complex reflection group. Theorem 1 ([7, 8]) The isomonodr omic deformations of type (B) connections (on rank thr ee vector bundles) ar e also contr olled by the P ainlev ´ e VI equation, and all P VI equa- tions arise in this way . Thus a solution to P VI can also be viewed as specifying an isomonodromic family of rank three connections. It turns out that the formulae to go from a P VI solution y ( t ) to such an isomonodromic family are more symmetrical than in the previous case (type (A) ) so we will recall them here. (For the analogous formulae for (A) see [31] and in Harnad’ s dual picture—the formula for which should be compared to that belo w—see [24] and also [43], which was kindly pointed out by the referee.) First the parameters: let λ i = T r ( B i ) for i = 1 , 2 , 3 and let µ i be the eigen v alues, in some order , of B 1 + B 2 + B 3 (which is minus the residue at infinity), so that ∑ λ i = ∑ µ i . Theorem 2 ([10]) If y ( t ) solves P ainlev ´ e VI with parameter s θ where θ 1 = λ 1 − µ 1 , θ 2 = λ 2 − µ 1 , θ 3 = λ 3 − µ 1 , θ 4 = µ 3 − µ 2 , and we define x ( t ) via x = 1 2  ( t − 1 ) y 0 − θ 1 y + y 0 − 1 − θ 2 y − t − t y 0 + θ 3 y − 1  then the family of logarithmic connections (B) will be isomonodr omic as t varies, wher e B 1 =   λ 1 b 12 b 13 0 0 0 0 0 0   , B 2 =   0 0 0 b 21 λ 2 b 23 0 0 0   , B 3 =   0 0 0 0 0 0 b 31 b 32 λ 3   b 12 = λ 1 − µ 3 y + ( µ 1 − xy )( y − 1 ) , b 32 = ( µ 2 − λ 2 − b 12 ) / t , b 13 = λ 1 t − µ 3 y + ( µ 1 − xy )( y − t ) , b 23 = ( µ 2 − λ 3 ) t − b 13 , b 21 = λ 2 + µ 3 ( y − t ) − µ 1 ( y − 1 ) + x ( y − t )( y − 1 ) t − 1 , b 31 = ( µ 2 − λ 1 − b 21 ) / t . 12 The implication of this for algebraic solutions should no w be clear: the monodromy of a P VI solution is also described by an action of the free group F 2 on (conjugacy classes of) triples of three dimensional complex reflections ( r 1 , r 2 , r 3 ) (with the same formula as before, just replace M i by r i ). Thus in this context the “ob vious” topological solutions (i.e finite F 2 or- bits) come from taking a triple of generating reflections of a finite complex reflection group in GL 3 ( C ) . Such finite complex reflection groups were classified in 1954 by Shephard and T odd [50] and apart from the familiar real orthogonal reflection groups there is an infinite family plus four exceptional complex groups, the Klein reflection group (of order 336, a two-fold cover of Klein’ s simple group isomorphic to PSL 2 ( F 7 )  → PGL 3 ( C ) ), two Hessian groups and the V alentiner group (of order 2160, a six fold cov er of A 6  → PGL 3 ( C ) ). The infinite family of groups and the two Hessian groups do not seem to lead to interesting ne w solutions, but by computing the F 2 orbits (determining the topology of Π ) it is easy to see that the Klein group yields a genus zero degree 7 solution and the V alentiner group has three inequi valent triples of generating reflections, each leading to genus one solutions with degrees 15 , 15 , 24 respectively . These are new solutions, previously undetected. (The 24 appearing here led to a certain amount of trepidation, gi ven that the 10 page elliptic solution of [21] had degree 18.) 4.1 Construction Of course finding the topological solution is not the same as finding an explicit isomon- odromic family of connections; one needs to solve a family of Riemann–Hilbert problems in verting the transcendental Riemann–Hilbert map for each value of t . (Indeed the author’ s original plan was to just prove the existence of new interesting solutions, in [7], but a cer- tain stubbornness, and some inspiration from reading about Klein’ s work finding explicit 3 × 3 matrices generating his simple group, con vinced us to go further .) The two main steps in the method we finally got to w ork are as follo ws. (This is a general- isation of the method used by Dubrovin–Mazzocco [21].) 1) Jimbo’ s asymptotic formulae. In [30] M. Jimbo found an exact formula for the leading asymptotics at t = 0 of the branch of the P VI solution y ( t ) corresponding to an y suf ficiently generic linear monodromy representation ( M 1 , M 2 , M 3 ) . (This formula was obtained by considering the degeneration of the isomonodromic family of connections (A) as t → 0; in the limit the four punctured sphere degenerates into a stable curve with two components, each with three marked points. The connections (A) degenerate into hypergeometric sys- tems on each component, with known monodromy . Since these are rigid it is easy to solve their Riemann–Hilbert problems explicitly and this giv es the leading asymptotics of the isomonodromic family and thus of the P VI solution.) 13 00 11 00 11 00 11 00 11 00 11 00 00 11 11 00 11 00 11 00 00 11 11 00 11 00 00 11 11 00 11 0 1 ∞ B Figure 1: Degeneration to two hyper geometric systems This is useful for us because, as Jimbo mentions, one may substitute the leading asymp- totics back into the P VI equation to get arbitrarily many terms of the precise asymptotic expansion of the solution at 0. If the solution is algebraic, then this is its Puiseux expan- sion, a suf ficient number of terms of which will determine the entire solution. It turns out there was a typo in [30], which meant the entire method did not work (indeed the fact it did not work led to the questioning of Jimbo’ s formula and hence the correction [8]). (Note the special parameters of [21] are not covered by Jimbo’ s result; rather [21] adapted the argument of [30] to their case.) 2) Relating (A) and (B). Since Jimbo’ s formula requires a monodromy representation of a connection of type (A) , and we are starting with a triple of 3 × 3 complex reflections (the monodromy representation of a connection of type (B) ), the second step is that we need to see how to go between these two pictures (on both the De Rham and Betti sides of the Riemann–Hilbert correspondence). This will be described in the following subsection. 4.2 Relating connections (A) and (B) W e wish to sketch ho w to con vert a connection (B) on a rank three vector bundle into a connection of the form (A) on a rank two b undle. On the other side of the Riemann–Hilbert correspondence this amounts to an F 2 -equi variant map from triples of complex reflections to triples of elements of G = SL 2 ( C ) (as in [8] Section 2). Of course the monodromy groups change in a highly nontrivial way under this procedure. For example the Klein reflection group becomes the triangle group ∆ 237 ⊂ G , which is an infinite group, and the V alentiner group becomes the binary icosahedral group (leading to an unexpected relation between A 6 and A 5 ). After this procedure was put on the arxi v ([8]) we learnt [17] that it is essentially a case 14 of N. Katz’ s middle con volution functor [32], although our construction using the comple x analytic Fourier –Laplace transform is different from that of Katz (using l-adic methods) and from the work of Dettweiler –Reiter [16]. The basic picture which emerges is as follows (see Diagram 1), and ought to be better kno wn. It was obtained essentially by a careful reading of the article [2] of Balser , Jurkat and Lutz, although the basic idea of relating irre gular and Fuchsian systems by the Laplace transform dates back to Birkhof f and Poincar ´ e. (Dubrovin [19, 20] used an orthogonal analogue in relation to Frobenius manifolds, also using [2]. Moreover the top triangle is essentially a case of ‘Harnad duality’ [24] so for n = 3 we kne w we would obtain all P VI equations.) Diagram 1 A ∈ gl n ( C ) d − ∑ A i z − a i d z d −  A 0 z 2 + A z  d z r 1 , . . . , r n r i = 1 + e i ⊗ α i ( u − , u + , h ) ∈ U − × U + × H GL n ( C ) H H H H H H j        A i = E i A ? Riemann–Hilbert ? Stokes -  Fourier –Laplace -  Killing–Coxeter ( α i ( e j )) = hu + − u − H H H H H H j r n · · · r 1        u − 1 − hu + The basic idea is to describe a transcendental map from from gl n ( C ) to GL n ( C ) in two dif ferent ways (the two paths down the left and the right from the top to the bottom of the diagram). Choose n distinct complex numbers a 1 , . . . , a n and define A 0 = diag ( a 1 , . . . , a n ) . Roughly speaking (on a dense open patch) the left-hand column arises by defining A i = E i A (setting to zero all but the i th row of A ) and constructing the logarithmic connection d − ∑ A i z − a i d z having rank one residues at each a i . Then taking the monodromy of this yields n complex reflections r i (and if bases of solutions are chosen carefully one can naturally define vectors 15 e i and one-forms α i such that r i = 1 + e i ⊗ α i and that the e i form a basis). Then the map to GL n ( C ) is giv en by tak en the product of r n . . . r 1 of these reflections, written in the e i basis. No w the ke y algebraic fact, which dates back at least to Killing [33] (see Coleman [15]), is that an y such product of complex reflection lies in the big cell of GL n ( C ) and so may be factored as the product of a lower triangular and an upper triangular matrix. W e write this product as u − 1 − hu + with u ± ∈ U ± the unipotent triangular subgroups, and h ∈ H diagonal: r n · · · r 2 r 1 = u − 1 − hu + . (2) Further , although this relation between the reflections and u ± looks to be highly nonlinear , one can relate them in an almost linear fashion: the matrix hu + − u − is the matrix with entries α i ( e j ) . On the other hand it turns out that the same map can be defined by taking the Stokes data of the irre gular connection d −  A 0 z 2 + A z  d z (indeed the map on the right-hand side generalises [6] to any complex reducti ve group G in place of GL n ( C ) , b ut only for GL n ( C ) is the alter - nati ve “logarithmic” vie wpoint a vailable). Thus u ± are also the tw o Stok es matrices of this irregular connection (the natural analogue of monodromy data for such connections); the exact definition is not important here. (The element h is the so-called formal monodromy , explicitly it is simply exp ( 2 π i Λ ) where Λ is the diagonal part of A .) The two connections are related (see [2]) by the F ourier–Laplace transform: this is more than just formal, and by relating bases of solutions on both sides the stated relation between the Stokes and mon- odromy data is obtained. (In both cases the resulting element of GL n ( C ) is the monodromy around z = ∞ in a suitable basis.) In summary we see that the “Betti” incarnation of the Fourier –Laplace transform is the relation of Killing–Coxeter . No w to apply this in the current context we consider the ef fect of adding a scalar λ to A ∈ gl n ( C ) . On the right-hand side this corresponds to tensoring the irregular connec- tion by the logarithmic connection d − λ d z / z on the trivial line bundle, and it follows [2] that the Stokes data is changed only by scaling h by s : = exp ( 2 π i λ ) , fixing u ± . On the logarithmic side this corresponds to a nontrivial con volution operation, changing the mon- odromy representation in a nontrivial way . Of course using the Killing–Coxeter identity we now see precisely how the complex reflections vary . (It is perhaps worth noting that this scalar shift is essentially the in verse of the spectral parameter introduced by Killing [33] p.20, appearing in the characteristic polynomial of the Killing–Coxeter matrix (2): det ( u − 1 − shu + − 1 ) = det ( shu + − u − ) .) If we set n = 3 then the logarithmic connections appearing are of the form (B) , upon taking a 1 , a 2 , a 3 = 0 , t , 1. Then we may choose the scalar shift such that the resulting element of GL 3 ( C ) has 1 as an eigen value. This implies that the connections are reducible and we can take the irreducible rank two sub- or quotient connection. Projecting to sl 2 gi ves the desired connection of type (A) (see [8]). (Note that there is a choice in volved here, of which eigen v alue to shift to 1.) 16 4.3 New solutions Thus in summary the procedure no w is as follo ws: take a triple of generating reflections of a finite complex reflection group in GL 3 ( C ) . Push it down to the 2 × 2 frame work using the scalar shift to obtain a triple ( M 1 , M 2 , M 3 ) of elements of SL 2 ( C ) in an isomorphic F 2 orbit. Apply Jimbo’ s formula to get the leading asymptotics of the corresponding P VI solutions at t = 0 on each branch (i.e. for each triple in the F 2 orbit). (Con verting the values which arise into exact algebraic numbers.) Substitute these leading terms back into P VI to obtain arbitrarily many terms of the Puiseux expansion at 0 of each solution branch. Use these expansions to determine the polynomial F ( y , t ) defining the solution (assuming it is algebraic). Find a parametrisation of the resulting algebraic curve (for example using M. v an Hoeij’ s wonderful Maple algebraic curves package). For e xample for the Klein complex reflection group of order 336 this works perfectly ([8]) and the resulting solution is: Klein solution θ = ( 2 , 2 , 2 , 4 ) / 7 y = −  5 s 2 − 8 s + 5   7 s 2 − 7 s + 4  s ( s − 2 ) ( s + 1 ) ( 2 s − 1 ) ( 4 s 2 − 7 s + 7 ) t =  7 s 2 − 7 s + 4  2 s 3 ( 4 s 2 − 7 s + 7 ) 2 which has 7 branches. One may of course no w substitute this back into the formula of Theorem 2 (with λ = ( 1 , 1 , 1 ) / 2 , µ = ( 3 , 5 , 13 ) / 14 ) to obtain an explicit family of loga- rithmic connections ha ving monodromy equal to the Klein reflection group generated by reflections (see [10] §3). When con verted to connections of type (A) these “Klein connections” hav e infinite (projec- ti ve) monodromy group equal to the triangle group ∆ 237 (cf. [11] Appendix B). On the other hand it turns out [9] that for the V alentiner connections, ev en though they are much trickier to construct directly , we can still compute immediately that they become connections of type (A) with binary icosahedral monodromy . They are also inequiv alent to those appear- ing in the w ork of Dubrovin–Mazzocco related to the real orthogonal icosahedral reflection group (which lead to unipotently generated monodromy with one choice of the scalar shift, but finite binary icosahedral monodromy with a dif ferent choice, cf. [9] Remark 16). Thus it seemed like a good idea to examine precisely what P VI solutions arise upon tak- ing arbitrary triples of generators ( M 1 , M 2 , M 3 ) of the binary icosahedral group. Thus we looked at all triples of generators and quotiented by the relation coming from the affine F 4 symmetries of P VI . The resulting table has 52 ro ws (which is quite small considering there are 26688 conjugacy classes of generating triples). The first 10 rows correspond to the 10 icosahedral rows of Schwarz’ s list and thus the projectiv e monodromy around one of the four punctures is the identity (these correspond to the P VI solution y = t ). The remaining ro ws are as in T able 2 (this is abridged from [9]). (Note that the right notion of equiv alence 17 in the linear nonrigid problem (A) seems to be the ‘geometric equiv alence’ of [9] section 4—-ho wev er this coincides with equi valence under the af fine F 4 W eyl group, in this case.) Degree Genus W alls T ype Degree Genus W alls T ype 11 2 0 2 b 2 c 2 32 10 0 3 d 4 12 2 0 2 b 2 d 2 33 12 0 0 a b c d 13 2 0 2 c 2 d 2 34 12 1 1 a b c 2 14 3 0 1 b c 2 d 35 12 1 1 a b d 2 15 3 0 1 b c d 2 36 12 1 1 b 2 c d 16 4 0 2 a c 3 37 15 1 2 b 3 c 17 4 0 2 a d 3 38 15 1 2 b 3 d 18 4 0 2 c 3 d 39 15 1 2 b 2 c 2 19 4 0 2 c d 3 40 15 1 2 b 2 d 2 20 5 0 1 b 2 c d 41 18 1 3 b 4 21 5 0 2 c 2 d 2 42 20 1 1 a b 2 c 22 6 0 1 b c 2 d 43 20 1 1 a b 2 d 23 6 0 1 b c d 2 44 20 1 3 a 2 c 2 24 8 0 1 a c 2 d 45 20 1 3 a 2 d 2 25 8 0 1 a c d 2 46 24 1 2 a b 3 26 9 1 2 b c 3 47 30 2 2 a 2 b c 27 9 1 2 b d 3 48 30 2 2 a 2 b d 28 10 0 2 a 2 c d 49 36 3 3 a 2 b 2 29 10 0 2 b 3 c 50 40 3 3 a 3 c 30 10 0 2 b 3 d 51 40 3 3 a 3 d 31 10 0 3 c 4 52 72 7 3 a 3 b T able 2: Icosahedral solutions 11 − 52 Thus there are lots of other icosahedral solutions the largest having genus 7 and 72 branches. (The column “T ype” indicates the set of conjugac y classes of local monodromy of the cor - responding connections of type (A) , as we marked on Schwarz’ s list. The column “W alls” indicates the number of reflection hyperplanes for the affine F 4 W eyl group that the so- lution’ s parameters θ lie on.) A few of these solutions had appeared before: those with degree less than 5 are simple deformations of pre vious solutions, solutions 21 and 26 are in Kitae v [36] and the Dubrovin–Mazzocco icosahedral solutions are equiv alent to those on ro ws 31 , 32 , 41. On the other hand the V alentiner solutions are quite far down the list on ro ws 37 , 38 and 46. The abov e method of constructing solutions using Jimbo’ s asymptotic formula applies only to sufficiently generic monodromy representations but it turns out that most of the rows of this table ha ve some representati ve (in their af fine F 4 orbit) to which Jimbo’ s formula maybe 18 applied (on ev ery branch). Thus we could start working down the list constructing new solutions. An initial goal was to get to solution 33: this solution purports to be on none of the reflection hyperplanes and the folklore was that all explicit solutions to Painle v ´ e equations must lie on some reflection hyperplane. The folklore was wrong: “Generic” solution, ro w 33, θ = ( 2 / 5 , 1 / 2 , 1 / 3 , 4 / 5 ) y = − 9 s ( s 2 + 1 )( 3 s − 4 )( 15 s 4 − 5 s 3 + 3 s 2 − 3 s + 2 ) ( 2 s − 1 ) 2 ( 9 s 2 + 4 )( 9 s 2 + 3 s + 10 ) t = 27 s 5 ( s 2 + 1 ) 2 ( 3 s − 4 ) 3 4 ( 2 s − 1 ) 3 ( 9 s 2 + 4 ) 2 . So far this looks to be the only example of a ‘classical’ solution of any of the Painle v ´ e equations that does not lie on a reflection hyperplane (of the full symmetry group). Apart from being in the interior of a W eyl alcove this solution is generic in another sense: a randomly chosen triple of generators of the binary icosahedral group is most likely to lead to it (more of the 26688 triples of generators correspond to this ro w than to any other). Notice also that this solution has type abcd ; there is one local monodromy in each of the four nontri vial conjugacy classes of A 5 . At this stage we were approaching solution 41 which we knew took 10 pages to write do wn. So we stopped and looked around to see if there were other interesting (ev en just topological) solutions. (The tetra/octahedral cases could all now be fully dealt with [11].) 5 Pullbacks In his book [39] on the icosahedron Klein showed that all second order Fuchsian differential equations with finite monodromy are (essentially) pullbacks of a hypergeometric equation along a rational map f : P 1 \ { k points } P 1 P 1 \ { 0 , 1 , ∞ } - Schwarz map y 1 / y 2 H H H H H H j f        / Γ (in variants) In particular ( k = 3) all the icosahedral entries on Schw arz’ s list, may be obtained by pulling back the“235” hyper geometric equation (on row VI of Schwarz’ s list). In our conte xt, an isomonodromic family of connections of type (A) amounts to a family of Fuchsian equations with 5 singularities (at 0 , t , 1 , ∞ plus an apparent singularity at another point y ). 3 Klein’ s theorem says each element of this family arises as the pullback of the 3 this is the same y appearing in P VI —i.e. the function y on the space of connections (A) is the position of the apparent singularity that appears when the connection is con verted into a Fuchsian equation [23]. 19 235 hypergeometric equation along a rational map, so the family corresponds to a family of rational maps. Thus finding a P VI solution corresponding to a family of connections (A) with finite mon- odromy amounts to giving a certain family of rational maps f : P 1 → P 1 . T o construct such P VI solutions one may try to find such families of rational maps, such that each map pulls back a hypergeometric equation to an equation with the right number of singular points— or to one that can be put in this form after using elementary transformations to remove extraneous apparent singularities. (This is not straightforward; for example giv en a finite monodromy representation of a connection (A) it is not immediate e ven what degree such a map f will hav e.) An important further observ ation (due to C. Doran [18] and A. Kitae v [35]) is that an y such family of rational maps will lead to algebraic solutions of Painle v ´ e VI regardless of whether or not the hypergeometric equation being pulled back has finite monodromy (provided the equation upstairs has the right number of poles); the algebraicity follo ws from that of the family of rational maps. Kitae v and Andreev [35, 1, 36] have used this to construct some P VI solutions, essentially by starting to enumerate all such rational maps (this leads to a fe w ne w solutions, but most in fact turn out to be equiv alent to each other or to ones previously constructed—see the summary at the end of this article). On the other hand, Doran had the idea that interesting P VI solutions should come from hy- pergeometric equations with interesting monodromy groups. Thus (amongst other things) [18] studied the possible hypergeometric equations with monodromy a hyperbolic arith- metic triangle gr oup which may be pulled back to yield P VI solutions. Indeed in [18] Corollary 4.6, Doran lists such possible triangle groups and the degrees and ramification indices of the corresponding rational maps f , although no ne w solutions were actually con- structed. W e picked up on this thread in [11] Section 5: it was found that all but one entry on Doran’ s list corresponded to a known e xplicit solution (although were perhaps unknown when [18] w as published). The remaining entry was for a f amily of de gree 10 rational maps f pulling back the 237 triangle group with ramification indices (partitions of 10): [ 2 , 2 , 2 , 2 , 2 ] , [ 3 , 3 , 3 , 1 ] , [ 7 , 1 , 1 , 1 ] ov er 0 , 1 , ∞ (where the hypergeometric system has projectiv e monodromy of orders 2 , 3 , 7 resp.), as well as minimal ramification [ 1 8 , 2 ] ov er another variable point. As explained in [11] one can get from here to a topological P VI solution by drawing a picture: we wish to find such a rational map f topologically—i.e. describe the topology of a branched cov er f : P 1 → P 1 with this ramification data. This may be done by playing “join the dots” (com- pletely in the spirit of Grothendieck’ s Dessins d’Enfants) and yields a co vering diagram as required. One diagram so obtained is sho wn in figure 2. (Note that the idea of drawing pictures such as figure 2 first appeared in Kitae v [36].) 20 2 2 2 2 2 1 3 3 1 1 1 1 2 P 1 2 3 7 7 · 3 1 · · f P 1 Figure 2: 237 degree 10 rational map f The upper copy of P 1 is thus divided into 10 connected components and f maps each component isomorphically onto the complement of the interv al drawn on the lo wer P 1 (the lines and the vertices upstairs are the preimages of the lines and vertices do wnstairs). In particular the diagram sho ws how loops upstairs map to words in the generators of the fundamental group π 1 ( P 1 \ { 0 , 1 , ∞ } ) do wnstairs. In this way we can compute by hand the monodromy of the equation upstairs obtained by pulling back a hypergeometric equation with monodromy ∆ 237 . This yields the triple: M 1 = caca − 1 c − 1 , M 2 = c , M 3 = c − 1 a − 1 cac (where a , b , c are lifts to SL 2 ( C ) of standard generators of ∆ 237 with cba = 1), which we kno w a priori liv es in a finite F 2 orbit. One finds immediately that the orbit through the conjugacy class of this triple has size 18 and constitutes a genus one, de gree 18 topological P VI solution. No w it turns out that Jimbo’ s formula may be applied to e very branch of this solution, and 21 proceeding as before we obtain the solution explicitly: Elliptic 237 solution θ = ( 2 / 7 , 2 / 7 , 2 / 7 , 1 / 3 ) y = 1 2 −  3 s 8 − 2 s 7 − 4 s 6 − 204 s 5 − 536 s 4 − 1738 s 3 − 5064 s 2 − 4808 s − 3199  u 4 ( s 6 + 196 s 3 + 189 s 2 + 756 s + 154 ) ( s 2 + s + 7 ) ( s + 1 ) t = 1 2 −  s 9 − 84 s 6 − 378 s 5 − 1512 s 4 − 5208 s 3 − 7236 s 2 − 8127 s − 784  u 432 s ( s + 1 ) 2 ( s 2 + s + 7 ) 2 where u 2 = s ( s 2 + s + 7 ) . (This solution, or rather an inequiv alent ‘Galois conjugate’ of it, has also been obtained independently by A. Kitaev [37] p.219 by directly computing such a family of rational maps—apparently also influenced by Doran’ s list.) 6 Final steps 6.1 Up to degr ee 24 W e now hav e an example of a degree 18 elliptic solution to P VI with a quite simple form. This leads immediately to the suspicion that the 10 page Dubrovin–Mazzocco solution is just written at a bad value of the parameters. Indeed using the method we have been ‘tweaking’ while working down the icosahedral table enables us to guess good a priori choices of the parameters θ within the af fine F 4 equi valence class for ro w 41 in T able 2 (i.e. so that the e xpression for the polynomial F will be ‘small’). Choosing such parameters and constructing the solution from scratch at those parameters yields: Theorem 3 ([9]) The Dubr ovin–Mazzocco icosahedral solution is equivalent to the solu- tion y = 1 2 − 8 s 7 − 28 s 6 + 75 s 5 + 31 s 4 − 269 s 3 + 318 s 2 − 166 s + 56 18 u ( s − 1 ) ( 3 s 3 − 4 s 2 + 4 s + 2 ) t = 1 2 + ( s + 1 )  32 ( s 8 + 1 ) − 320 ( s 7 + s ) + 1112 ( s 6 + s 2 ) − 2420 ( s 5 + s 3 ) + 3167 s 4  54 u 3 s ( s − 1 ) on the elliptic curve u 2 = s ( 8 s 2 − 11 s + 8 ) with θ = ( 1 , 1 , 1 , 1 ) / 3 . In particular this elliptic curve is birational to that defined by the ten page polynomial. 22 Substituting this into the formula of Theorem 2 with λ = ( 1 , 1 , 1 ) / 2 , µ = ( 1 , 3 , 5 ) / 6 now gi ves the third (and trickiest) f amily of connections of type (B) with monodromy the icosa- hedral reflection group. This can be pushed further with more tweaking to get up to degree 24 (row 46 in T able 2) i.e. to obtain the largest V alentiner solution [9] (the main further tricks used are described in [11] Appendix C). In particular this finishes the construction of all elliptic icosahedral solutions. Intriguingly , one finds that the resulting elliptic icosahedral Painle v ´ e curves Π become singular only on reduction modulo the primes 2, 3 and 5 (e xcept for ro ws 44 , 45— we will see another reason in the following subsection that these are abnormal). Similarly the elliptic Painle v ´ e curve related to the 237 triangle group becomes singular only on re- duction modulo 2, 3 and 7. 6.2 Quadratic/Landen/F olding transformations No w the happy fact is that the remaining icosahedral solutions may be obtained from earlier solutions by a trick, first introduced in the conte xt of P VI by Kitae v [34] and a simpler equi valent form was found by Ramani et al [47]. Manin [40] refers to some equiv alent transformations as Landen transformations (Landen has clear precedence since the original Landen transformations were rediscov ered by Gauss!). Tsuda et al [52] call them folding transformations. In any case the basic idea is simple: if one has a connection (A) with two local projectiv e monodromies of order two (say at 0 , ∞ ) then one can pull it back along the map z 7→ z 2 and obtain a connection with only apparent singularities at 0 , ∞ (which can be remov ed) and four genuine singularities. This can be normalised into the form (A) , and the k ey point is that this works in families and maps isomonodromic deformations of the original con- nections to isomonodromic deformations of the resulting connections—i.e. it transforms certain solutions of P VI into different, generally inequi valent, solutions. Of course this is not a genuine symmetry of P VI since special parameters are required, but it is precisely what is needed to construct the remaining solutions. Indeed observe that each of the rows of the icosahedral table with degree greater than 24 hav e type a 2 ξη for some ξ , η ∈ { a , b , c , d } —i.e. they hav e two projectiv e monodromies of order two. Pulling back along the squaring map will transform the corresponding con- nections into connections of type ξ 2 η 2 . It turns out (in this icosahedral case) the corre- sponding P VI solutions hav e half the degree, and we obtain an algebraic relation between the solutions. This program is carried out in [12] and the remaining icosahedral solutions are obtained. See also Kitae v–V id ¯ unas [38]. (Notice also that the elliptic solutions on ro ws 44 , 45 are related in this way to earlier solutions.) For example in [12] we found a relati vely simple explicit equation for the genus 7 algebraic curve naturally attached to the icosahe- dron, on which the largest (degree 72) icosahedral solution is defined: it may be modelled 23 as the plane octic with af fine equation 9 ( p 6 q 2 + p 2 q 6 ) + 18 p 4 q 4 + 4 ( p 6 + q 6 ) + 26 ( p 4 q 2 + p 2 q 4 ) + 8 ( p 4 + q 4 ) + 57 p 2 q 2 + 20 ( p 2 + q 2 ) + 16 = 0 . The genus se ven icosahedral Painle v ´ e curve 7 Conclusion Thus in conclusion we ha ve filled in a number of rows of what could be called the nonlinear Schwarz’ s list . Whether or not there will be other ro ws remains to be seen. So far this list of kno wn algebraic solutions to P VI takes the follo wing shape (we will use the letters d and g to denote the de gree and genus of solutions, and consider solutions up to equi valence under Okamoto’ s affine F 4 symmetry group. Some non-trivial work has been done to establish which of the published solutions are equiv alent to each other and which were genuinely ne w): First there are the rational solutions ( d = 1), studied by Mazzocco [42] and Y uan–Li [54], which fit into the set of Riccati solutions classified by W atanabe [53]. (Be ware that ‘rational’ here means the solution is a rational function of t , which implies, but is by no means equi valent to, ha ving a rational parameterisation.) Then there are three continuous families of solutions g = 0 , d = 2 , 3 , 4. The degree two family is y = √ t which, as one may readily verify , solves P VI for a family of possible parameter values. Similarly the degree 3 tetrahedral solution, and the degree 4 octahedral and dihedral solutions (of [19, 25, 27]) fit into such families, as discussed in [9, 3, 13]. In general in such a f amily y ( t ) may depend on the parameters of the family . Ben Hamed and Gavrilo v [3] sho wed that any family with y ( t ) not depending on the parameters is equi valent to one of the abo ve cases and recently Cantat and Loray [13] sho wed that any solution with 2 , 3 or 4 branches is in such family . Next there is one discrete family ( d , g unbounded, θ = ( 0 , 0 , 0 , 1 ) ∼ ( 1 , 1 , 1 , 1 ) / 2). Indeed this P VI equation was solved completely by Picard [46] p.299, R. Fuchs [23] and in a dif- ferent way by Hitchin [26]. Algebraic (determinantal) formulae for the algebraic solutions amongst these appear in [25], using links with the Poncelet problem—in this framework they are dihedral solutions (controlling connections of type (A) with binary dihedral mon- odromy). Finally there are 45 exceptional solutions , which collapse do wn to 30 if we identify so- lutions related by quadratic transformations. The possible genera are 0 , 1 , 2 , 3 , 7 and the 24 highest degree is 72. Of these 30 solutions 7 hav e pre viously appeared: one is due to Dubrovin [19], two to Dubro vin–Mazzocco [21] and four to Kitae v (three in [36], plus—in [37]—a Galois conjugate of the elliptic 237 solution already mentioned). T wo of these exceptional solutions are octahedral, one is the Klein solution, three are the elliptic 237 solution (and its two Galois conjugates) and the remaining twenty-four are icosahedral. Refer ences [1] F . V . Andree v and A. V . Kitae v , T ransformations RS 2 4 ( 3 ) of the ranks ≤ 4 and alge- braic solutions of the sixth Painlev ´ e equation , Comm. Math. Phys. 228 (2002), no. 1, 151–176. [2] W . Balser , W .B. Jurkat, and D.A. Lutz, On the r eduction of connection pr oblems for differ ential equations with an irr e gular singularity to ones with only r e gular singular - ities, I. , SIAM J. Math. Anal. 12 (1981), no. 5, 691–721. [3] B. Ben Hamed and L. Gavrilov , F amilies of Painlev ´ e VI equations having a common solution , Int. Math. Res. Not. 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Math. 54 (2002), no. 3, 648–670. ´ Ecole Normale Sup ´ erieure et CNRS, 45 rue d’Ulm, 75005 Paris, France www .dma.ens.fr/ ∼ boalch boalch@dma.ens.fr 28