Painlevé VI equations with algebraic solutions and family of curves 1

Transcription

1 Painlevé VI equations with algebraic solutions and famil of curves Hossein Movasati Stefan Reiter Instituto de Matemática Pura e Aplicada IMPA Estrada Dona Castorina Rio de Janeiro RJ Brazil hossein@impa.br reiter@impa.br Abstract In families of Painlevé VI differential equations having common algebraic solutions we classif all the members which come from geometr i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families of algebraic varieties. In our case we have one famil with zero dimensional fibers and all others are families of curves. We use the classification of families of elliptic curves with four singular fibers done b Herfurtner in 99 and generalize the results of Doran in 00 and Ben Hamed and Gavrilov in 005. Introduction Along the solutions of the sixth Painlevé differential equation written in the vector field form ( P V I θ : K µ λ K λ µ + t in C with coordinates (λ µ t where t(t K = λ(λ (λ tµ (θ (λ (λ t+θ λ(λ t+(θ λ(λ µ+κ(λ t κ = 4 (( θ i θ4 and θ = (θ θ θ θ 4 is a fixed multi-parameter the linear differential equation ( + p (z + p (z = 0 p (z := p (z := θ z t + θ + θ z z z λ κ t(t K λ(λ µ + z(z z(z (z t z(z (z λ is isomonodromic i.e. its monodrom group representation is constant. Recentl there have been works on algebraic solutions of ( using linear differential equations coming from geometr i.e. those who are Picard-Fuchs equations associated to families of varieties. Linear equations ( with finite monodrom come automaticall from geometr and this is the origin of man algebraic solutions known until now (see [] and the references therein. Doran in [6] took 5 deformable families of elliptic curves with non-constant j Math. classification: 4M55 5Q5 Kewords: Painlevé sixth equation Okamoto transformation monodrom convolution

2 invariant along the deformation parameter and with exactl four singular fibers which were alread classified b Herfurtner in [0] and obtained algebraic solutions for ( (see Table column and row -6 for a = c =. In general one can obtain such algebraic solutions b taking pull-backs of the Gauss hpergeometric equation (see [] and the references therein. Ben Hamed and Gavrilov in [] took zero dimensional families of three points varieties constructed directl from the Herfurtner list and the obtained the same algebraic solutions in the (t λ-space but for different parameters θ. The have also noticed that in the parameter space of Painlevé equations ( the points obtained b them and Doran lie in families with algebraic solutions whose projections in the (t λ space is independent of the parameter of the famil. Then b a straightforward calculation the showed that up to the Okamoto transformations corresponding to the Möbius transformation of P such families of Painlevé equations are given b the first and second column of Table. The main result of this article is to classif the members of such families which come from geometr (third column of Table. Table : Algebraic solutions of families of the sixth Painlevé equation Algebraic solution (θ θ θ θ 4 famil of algebraic varieties {λ = t = 0} {λ = t = } (0 c = x a ( x b (z x +a c dx c a b b a λ = ( a+ a zero dimensional varieties a+c b t = b ( µ = a c+ b (a + c λ = b t = b (c a + c = (4x g x + g c (x + g /4 a dx µ = a c+ b c a + c g = 4( z + z g = b z 8 z z 8 z b = 4 (b + b + z = b z λ = b b t = b+ b 4 +b (a (a = (4x g x g a dx a a g = z ( z + b z + µ = ( a+b (b+ (b+ g = 4 z ( z + ( b + z + 6 b z + b = (b + b z = b +b z λ = b +b +b+ b +b 5b+ t = b4 6b 8b b 4 6b +8b (a = (4x g x g a dx a a g = z ( z + b g = z 5 ( z + b = µ = (a (b (b+ b 4(b+ b + + z = b b +b λ = b 4 b 4 6b t = b +8 b 6 6b (a 4 = (4x g x g a dx a g = z ( z + b b b +b 9 z a µ = ( a+b (b +(b 6 (b g = z 4 ( z + b z + b = 4 (b + b z = b b z Theorem. In the columns and of Table the corresponding linear differential equations ( come from geometr if and onl the exponent parameters a and c in column (and b in the first row are rational numbers. The corresponding famil of algebraic varieties and differential form are listed in column. Moreover the second famil (together with its algebraic solution is Okamoto equivalent to the third famil and the fourth famil is Okamoto equivalent to the fifth famil. Both Okamoto transformations are equivalent to the middle convolution of the corresponding Fuchsian sstems. In the first row of Table the corresponding linear differential equation ( is the Gauss hpergeometric equation and the geometric interpretation is classical and it has nice applications in the theor of the special values of Gauss hpergeometric functions (see [4]. Note that in this case the projection of the corresponding algebraic curve in the (λ µ t space into the (λ t space is just the zero dimensional variet {(0 0 ( }.

3 In all other cases it is a curve in the (λ t space. In the literature one finds mainl the equations of such curves. Note also that the parametrization of the algebraic solutions in column is different from the parametrization of the solutions of the vector field (. Let us put the parameters a and c in column and rows equal to. We obtain families of elliptic curves = 4x g (z bx g (z b with exactl four singular fibers and with j-invariant depending on an extra parameter b. These are exactl four families of the five Herfurtner families. The missing famil in the Herfurtner list is the one given b g = (z (z b g = (z (z b 4 (z + b. The corresponding famil of curves = (4x g x g ã gives us the Painlevé equation and its algebraic solution in Table row b setting c = 5 6 a = ã. Note that in this case we find two apparentl different geometric interpretations for the same Painlevé equation. Note also that in the mentioned five families of elliptic curves just for the famil used in row the polnomial 4x g (z bx g (z b is reducible in x. In row for a and c rational numbers we have shown that the monodrom group of the linear equation ( is a dihedral group and so it is finite. Also the other families are related via the middle convolution to (third order differential equations whose monodrom groups are finite imprimitive reflection groups for rational parameters. In [] Boalch started with third order differential equations to obtain (via a construction equivalent to the middle convolution algebraic solutions and the parameters of the corresponding Painlevé VI differential equations. Recentl Cantat and Lora showed in [4 Prop. 5.4] that an algebraic solution of a Painlevé VI differential equation having degree or 4 belongs (up to Okamoto transformation to one of the families in Table. Using Singular (see [8] for each linear equation ( we have calculated the corresponding Fuchsian sstem in the Schlesinger form and the Okamoto transformation of the algebraic solutions in Table corresponding to exchanging the role of the first and second coordinates of the sstem. The details of the calculation is explained for the example in row of Table and for the others the reader is referred to the home-page of the first author. Let us explain the content of each section. In we introduce sstems of linear differential equations in two and three variables. Pulling these back we get Fuchsian sstems with four singularities. In we recall some well-known facts about linear differential equations and in 4 we explain how to calculate Fuchsian sstems in Schlesinger form associated to families of curves in Table column. Such calculations are explained for the third row of Table in 5. In 6 we recall some basic facts about the middle convolution and in 7 we show that the algebraic solution in row of Table can be obtained via the middle convolution of the Fuchsian sstem corresponding to the algebraic solution in row of Table. In 8 we show that the Fuchsian sstem associated to the Painlevé equation in row of Table has finite monodrom if a c Q and so it comes from variation of zero dimensional varieties. Further we point out how all the other Picard-Fuchs differential equations are related via the middle convolution to those with finite monodrom. Finall in 9 we show that the middle convolution relates the results in Table row 4 and row 5. The second author gratefull acknowledges financial support b the CNPq and thanks IMPA for its hospitalit. The authors thank L. Gavrilov for useful discussions in the initial steps of the present article.

4 Linear sstems in two variables For a b c C fixed we consider the following famil of transcendent curves: ( E : = f(x f(x := (t t ( a c (t t ( a b (t t ( b c (x t a (x t b (x t c Here t = (t t t is a parameter in T := {t C (t t (t t (t t 0}. We distinguish three not necessaril closed paths in E. In the x-plane let δ i i = be the straight path connecting t i+ to t i i = (b definition t 4 := t and t 0 = t. There are man paths in E which are mapped to δ i under the projection on x. We choose one of them and call it δ i. We can make our choices so that δ + δ + δ is a limit of a closed and homotopic-to-zero path in E. For instance we can take the path δ i s in such a wa that the triangle formed b them has almost zero area. Now we have the integral (4 δ p(xdx = δ p(xdx f(x p C[x] where δ is one of the paths explained above. B a linear change in the variable x such integrals can be written in terms of the Gauss hpergeometric function (see []. Another wa to stud the integrals (4 is b using Pochhammer ccles. For simplicit we explain it for the pairs (t t. The Pochhammer ccle associated to the points t t C and the path δ is the commutator α = [γ γ ] = γ γ γ γ where γ is a loop along δ starting and ending at some point in the middle of δ which encircles t once anti-clockwise and γ is a similar loop with respect to t. It is eas to see that the ccle α lifts up to a closed path α in E t and if a b Z then p(xdx α = ( e πia ( e πib α p(x f(x dx. (see [] Proposition..7. For a fixed a T the period map is given b: ( (5 pm : (T a GL( C t dx δ xdx δ dx δ xdx δ where (T a means a small neighborhood of a in T. The map pm can be extended along an path in T with the starting point a. The period map pm is a fundamental sstem for the linear differential equation dy = AY in C where (6 A = (t t (t t (b + c t + (a + c t +! (a + b t a b c + at t + (b t t + (c t t (b + c t (a + c t (a + b t dt +( dt + ( dt 4

5 The matrix coefficient of dt (resp dt is obtained b permutation of t with t and a with b (resp. t with t and a with c in the matrix coefficient of dt written above. Now for the multi-valued function we have the sstem = (7t t ( a (4x t x t a (( (7 A = 7t 4 t 7at + 8t t 9 4 at t + 4 t t 4 t ( 9 t 8at t dt and for we have at t 9 t dt + = (t + t ( a c (t 6t ( c (4x t x + t c (x + 4 t a (8 A = (t 6t (t + t (( (6at t 6ct t t + 5t t ( 48at 96ct + 96t (at ct t + t t 4t ( 6at t + 6ct t + t 5t dt t + ( ( at + ct + t + 8t (4at + 48ct 48t ( 6at t + ct t + t t (at ct t 8t dt The calculation of the matrix A for the elliptic case a = b = c = is classical and goes back to Griffiths [9]. In fact the matrix A in (6 can be also calculated from the well-known Fuchsian sstem for hpergeometric functions i.e. for the case t = 0 t = t = t. The algorithms for calculating such a matrix are explained in [] and the implementation of such algorithms in a computer can be found in the first author s home-page. Review of Fuchsian differential equations Here we collect some basic facts about Fuchsian differential equations which can all be found in []. Let D = d dz and (9 D + p (zd + p (z = 0 be a Fuchsian differential equation with regular singularities at t... t m C and t m+ =. Then b [ Prop. 4.] p (z = m a i z t i p (z = m b m i (z t i + c i (z t i a i b i c i C where p (z m (z t i is a polnomial in C[z] of degree at most (m i.e. c i = 0. The following table of the regular singular points t... t m+ and the exponents s i s i at t i is called the Riemann scheme of the differential equation t... t m+ s... s m+ s... s m+ 5

6 j= si j where the sum of all exponents satisfies the Fuchs relation m+ = m. The exponents at the singularit t i i =... m are the roots of s(s + a i s + b i = 0 (see [ p. 70] and at t m+ the exponents satisf m m m s(s + ( a i s + ( b i + ( c i t i = 0. The second order Fuchsian differential equation (9 can be transformed into SL-form b substituting b f where 0 f satisfies Df = p (zf. Then b [ p. 66] (0 D = p(z p(z = p (z + 4 p (z + Dp (z. From a two dimensional sstem DY = QY Q = (q ij (not necessaril a Fuchsian sstem we obtain the second order differential equation (9 for the first coordinate of Y = ( tr (see [ Lemma 6..] with ( p (z = D log(q (z Tr(Q p (z = det(q(z Dq + q D log q. If λ is a zero of q of order r and λ {t... t m } then z = λ is an apparent singular point with the exponents 0 and r + (see [ Lemma 6..]. 4 The algorithm In this section we explain how to obtain the algebraic solutions of the Painlevé VI differential equation starting from the families of curves in Table column row which are constructed directl from the Herfurtner list of families of elliptic curves (see [0]. For the convenience of the reader we have listed the five families which we need: Table : List of deformable families of elliptic curves = 4x g x g with four singular fibers and with non constant j invariant along the deformation name deformation g = (z (z b g = (z (z b 4 (z + b g = z (z + bz + g = 4z (z + bz + bz + g = z (z + bz + g = 4z (z + (b + z + 6bz + 4 g = z (z + b g = z 5 (z + 5 g = z (z + b g = z 4 (z + bz + Let us consider f(x = 4x g x g where g g correspond to one of the five families of elliptic curves with parameter b in the Herfurtner list. Since the roots of the discriminant = 7g g of f which will be the singular points of the corresponding differential equation for dx δ are not all rational functions in b we substitute b b a suitable rational function in b such that the transformed roots are rational in b. Such substitutions are done b the equalities b = in Table column. In the next step we check whether the polnomial f(x factorizes over Q(b z. It turns out that this onl happens for second famil. In that case we have ( f(x = (4x g x + g (x + g 4. Substituting g and g in the list (resp. 4 5 of Table for t and t in (8 (resp. (7 we obtain a sstem DY = AY. We can now compute the second order differential 6

7 dx equation satisfied b δ where δ is a Pochhammer ccle using the formula ( and (0. It alwas turns out that (9 has four singularities at t t = 0 t and t 4 = and apparent singularit at λ with exponents 0 and. Hence the SL-form can be written as follows ([ p. 7] p(z = a i (z t i + a 4 z(z t + t (t t /t L z(z t (z t + 4 (z λ λ(λ t /t ν z(z t (z λ where a i = 4 (θ i i = a 4 = 4 ( θ i θ 4 a L = t ((p(z (z t (z t z=t ν = t ((p(z (z λ 4(z λ z=λ. The corresponding Riemann-scheme is t t t λ θ θ θ θ 4 The sstem +θ +θ +θ +θ 4. ( DY = QY Q = Q k k= z t k Q k = Q k (t = (qij k Mat n(c k = of Schlesinger tpe with the 4 regular singularities at t t = 0 t t 4 = can now be determined: We can assume that θ i and 0 are the eigenvalues of Q i and that (if θ 4 Q i (t = ( α 0 0 α + θ 4 α + 4 θ i =. Further we normalize the singularities via the Möbius transformation z z t to t = t t 0 and. (Note that also the apparent singularit is transformed to λ t which we will also denote again b λ. Then b [ Prop. 6..] the matrices Q i i = can be expressed as follows: ( (4 Q i = M i (W i W M i (W i W (M i (W W i + θ i θ i M i (W i W i =... M i W i i = λ t t(t λ(λ (µ + α λ (θ 4 W = W i(m i W i θ i i = λ t (λ t(λ (µ + α λ tα λ µ = ν θ i λ t i i = λ t λ(λ t(µ + α λ α = ( 4 θ i. 7

8 Note that a zero of Q has order r = which is the apparent singularit λ for the first coordinate. If we compute the differential equation for the second coordinate or equivalentl if we compute the differential equation for the first coordinate of the transformed sstem DY = T QT Y T = ( 0 0 then (θ... θ 4 is transformed to (θ... θ θ 4. 5 An example We demonstrate how the algorithm in 4 ields the results for the third row of of Table. We start with the second famil in Table. The roots of the discriminant of f are 0 ω ω ω = (b ± b b Solving the diophantic equation α α = β we substitute b b 4 (b + b + in g and g. Thus the new roots are t = b t = 0 t = b. Since f(x = 4x g x g factorizes we get ( with the new coefficients g := 4(z + z g := 9b z 8bz 4 +bz 8bz 9z b Substituting g and g for t and t in (8 we obtain a sstem which does not fit in our paper and we do not write it here. Computing the SL-form (0 of the differential equation for the first coordinate of the sstem we obtain the following parameters: (note that we have normalized the singularities via the Möbius transformation z z t (t t t = (t 0 (θ θ θ θ 4 = (c a + c c a + c The sstem ( using (4 reads: λ = b t = b µ = a c+ b ν = 4b. q = q = q = (ba + a + bc b (a + c 4b(a + c b(b (b (ba + a b + c ( ba a bc + b + (a + c (a 6b(a + c q = q + (c q = ( b + b (a + c (a 4b(a + c q = b 8

9 q = (b (( a c(a (b + + ( a 4ac + 6a 4c + 8c 4b(a + c (a 6b(a + c q = q + (a c q = (ba + a b + c (a + c 4(a + c q = b q = (b (ba + a b + c (ba + a + bc b (a + c (a 6(a + c q = q + (c and (Q + Q + Q = ( (a + c 0 0 c. The apparent singularit of the second coordinate is the zero of Q : λ = b + 6 Middle convolution ( a c + 4(b + b (a (a + c (b + + (a + ca 5a + c 6c + 6b. For the convenience of the reader we give here a short review of the middle convolution for Fuchsian sstems (see [5]. Let f(z = (f (z... f n (z tr be a solution of the Fuchsian sstem DY = AY = r A i z t i Y A i Mat n (C and [t i ] = γt i γ γ ti γ be a Pochhammer ccle around t i and. Then the Euler transform of f(z with respect to µ C and [t i ] [t i ] [t i ] f(z( zµ dz z t. dz f(z( zµ z t r i =... r is a solution of the (Okubo- sstem A... A r (I nr T DY = BY := (. + µ I nr Y T = diag(t I n... t r I n A... A r DY = r B i t i Y B i Mat nr (C 9

10 where I n is the identit n n matrix. In general this sstem is not irreducible and has the following two B... B r -invariant subspaces: k = r ker(a i l = ker(b = (v... v tr v ker( r A i + µi n. Factoring out this subspace we obtain a Fuchsian sstem in dimension m m = nr r r dim(ker A i dim(ker( A i + µi n. Let M i be the monodrom of the sstem DY = AY at t i. Then if the sstem is irreducible and rk( rk(a i = rk(m i I n i =... r r A i + µi n = rk(m M r λ I n λ = e πiµ then the factor sstem is again irreducible and it is called the middle convolution of DY = AY with µ. Thus if we start with a two dimensional sstem in Schlesinger form ( where the parameters are (θ... θ 4 then there are in general two possibilities to get again a new two dimensional sstem ( via the middle convolution. We can appl the middle convolution either with µ = α or with µ = α + θ 4 and diagonalize the residue matrix at to ( α 0 0 α + θ. 4 This changes the parameters as follows: ( θ... θ 4 = (θ + µ... θ + µ θ 4 µ + α α = µ. Note that it is possible to write down also the transformation for the apparent singularit (see [7 Section 5] which is also known as an Okamoto transformation. 7 Application of the middle convolution I In this section we show that the algebraic solution in row of Table can be obtained via the middle convolution of the Schlesinger sstem corresponding to the algebraic solution in row of Table. We continue with our example b determining the middle convolution of DY = QY with µ = c. Hence for the (Okubo- sstem (zi 6 T DY = BY DY = B i t i Y B i Mat 6 (C we get B = Q c I Q Q Q Q c I Q Q Q Q c I and T = diag(t t 0 0. The B B B invariant subspace has dimension 4 since q i ker Q i = ( q i l = (0 0 0 tr. 0

11 In order to get a dimensional factor sstem we set q q q S = 0 q q q 0 0 Thus D(SY = SB i S z t i (SY = à (SY and we get the dimensional factor sstem DY = ÃY = sstem into Schlesinger form via à i z t i Y. Transforming this ( Y SY ( S = Ãi + c I ( Ãi c I a I + I ( Ãi + c I ( Ãi c I a I + I where the columns of S consist of eigenvectors of Ãi with respect to the eigenvalues c and c + a we obtain finall (5 DY = AY A = A z b + A z + A z A = ( ab a bc + b + ab + a + bc b 4b ab a bc + b + ab + a + bc b A = ( ab + ab + a b b ab a b + 4b ab + a + b ab + ab a + b b + A = ( ab a + b c + ab a + b c + 4 ab + a b + c ab + a b + c and (A + A + A = ( c 0 0 (a + c. Computing the entries of A we get A = 4cz + (( a + c(b + + ( abz + (a (b + b 4z(z (z b A = (b ((a + c z + b(a 4z(z (z b A = (b (( a cz + (b ab 4z(z (z b A = 4(a + c z + ((5 a c(b + + (a bz + ( a(b b 4z(z (z b.

12 Thus the parameters are (θ... θ 4 = ( / a / (a + c. The apparent singularit λ for the first resp. λ for the second coordinate is Since t = b we obtain the relation a + λ = ( a + c b λ a + = ( a + c b. θ 4 λ = t θ ( θ 4 λ = t θ. Computing the SL-form for the first coordinate (after transforming Y (z / (z t / Y which changes the parameters (θ... θ 4 (but not the apparent singularities to (/ a / (a + c we get a simple formula for µ 8 Monodrom µ = a c +. b In this section we show that (5 arises also as a pullback of hpergeometric differential equations. B determining the monodrom group it turns out that this monodrom group is finite and therefore b a well known result of Klein the claim follows. Further we indicate that all the other families of Picard-Fuchs equations are related to those with finite monodrom. Proposition. The projective monodrom group of (5 is (up to conjugation contained in the orthogonal group GO (C. Moreover if the parameters a and c are rational numbers i.e. if (5 is a Picard-Fuchs equation then the monodrom group is even finite i.e. a dihedral group. Proof. To prove this statement let M t M 0 M denote the monodrom at t 0 and. If the parameter b tends to we see that the sstem becomes reducible and monodrom group is abelian: A ( a cz+ z z 0 0 a+c z Hence M t M and M 0 commute and are diagonal matrices. Since M t and M are reflections the group generated b them is an orthogonal group. Hence M t and M normalize also M 0 and the claim follows. Appling the Möbius transformation z z that permutes the residue matrices Q i of the Schlesinger sstem corresponding to row 5 (resp. row 6 column in Table and scaling the new Q we get the following pairs of eigenvalues for Q Q Q (Q +Q +Q resp. (a 0 (a 0 (a. 0 ( a a (a 0 ( a + 0 (a 0 (.

13 The middle convolution with µ = (a ields a three dimensional Fuchsian sstem with the following triples of eigenvalues of the residue matrices resp. ( 0 0 ( 0 0 ( 0 0 ( a a a ( 0 0 ( a ( 0 0 ( + a + a a. Using the explicit construction for the middle convolution and similar arguments as in the above Proposition one easil sees that the monodrom groups of these third order differential equations are finite imprimitive reflection groups contained in T S where T denotes the group of diagonal matrices. In the next section we show that in Table row 4 and row 5 are also related via the middle convolution. Hence all the families of Picard- Fuchs equations corresponding to Table are related to those with finite monodrom. Since the corresponding Picard-Fuchs differential equation of row in Table is the hpergeometric one it is well known that it is obtained via the convolution of a one dimensional differential equation with finite monodrom. 9 Application of the middle convolution II As in the previous example we show that the middle convolution relates the results in Table row 4 and row 5. The sstem ( in Schlesinger form corresponding to the differential equation satisfied b δ dx where is from row 4 in Table reads: q = (8a b + 8a b + 8a ab 0ab 48ab 6a + b + b + 4b + 6 8ab + 8a 7b 7 q = b4 b b q = q q q q = (a q q = (a (b + ( 6a + 7b + (b 8ab 7b q = b + b q = q q q q = (a q q = (8a b + 8a b + 8a b 6ab 48ab 0ab a + 6b + 4b + b + 8ab + 8ab 7b 7b q = b + b b q = q q q q = (a q

14 (Q + Q + Q = ( a a +. Appling the middle convolution to DY = QY with µ = (a and transforming the dimensional factor sstem into Schlesinger form with singularities at t 0 and we get the sstem ( with q = q = (a (b + ( 6a + (b + 9(4a (b + b + (7a 8(b + q = (a ((6a 4b + ( 6a + 60a 4b + (4a b 5( b + (6a (b + 9(4a (b + q = ( a + q q = ( a + (b + ( 6a + 4b + (b + b + 9(4a b q = (b + b + 8(4a b q = ((4 6ab4 + (6a 54a + 0b + (6a 96a + b + (6a 54a + 0b + 4 6a 9(4a b q = q q = q = (a ( b + (6a 4b (a (( 6a + b + (b + 9(4a b (b + b + 8(4a b (b + q = (a ( 5b + (4a b + ( 6a + 60a 4b + (6a 4((6a b (b + 9(4a b (b + q = ( a + q ( a 0 (Q + Q + Q = 0 a = ( a 0 0 a + ( a +. We obtain the parameters (θ... θ 4 = ( a + a + a +. The apparent singularit for the first coordinate is Using that t = b+ b 4 +b λ = b + b + b + b. we see that λ and t satisf λ 4 tλ λ + 6tλ t λ tλ + t t + t = 0. 4

15 Since this is also the relation for from Table row 5 the claim follows. References t = b4 6b 8b b 4 6b + 8b λ = b + b + b + b + b 5b + [] B. Ben Hamed and L. Gavrilov. Families of Painlevé VI equations having a common solution. Int. Math. Res. Not. (60: [] P. Boalch. Painlev equations and complex reflections. Ann. Inst. Fourier 5 no [] P. Boalch. The fift-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596: [4] S. Cantat and F. Lora. Holomorphic dnamics Painlev VI equation and Character Varieties. arxiv :07.579v [5] M. Dettweiler and S. Reiter. Middle convolution of Fuchsian sstems and the construction of rigid differential sstems. J. Algebra 8 no [6] C. F. Doran. Algebraic and geometric isomonodromic deformations. J. Differential Geom. 59(: [7] Y. Haraoka and G. Filipuk Middle convolution and deformation for Fuchsian sstems. J. Lond. Math. Soc. ( 76 no [8] G.-M. Greuel G. Pfister and H. Schönemann. Singular.0. A Computer Algebra Sstem for Polnomial Computations Centre for Computer Algebra Universit of Kaiserslautern [9] P. A. Griffiths. The residue calculus and some transcendental results in algebraic geometr. I II. Proc. Nat. Acad. Sci. U.S.A. 55:0 09; [0] S. Herfurtner. Elliptic surfaces with four singular fibres. Math. Ann. 9(: [] K. Iwasaki H. Kimura S. Shimomura and M. Yoshida. From Gauss to Painlevé. Aspects of Mathematics E6. Friedr. Vieweg & Sohn Braunschweig 99. A modern theor of special functions. [] A. V. Kitaev. Grothendieck s dessins d enfants their deformations and algebraic solutions of the sixth Painlevé and Gauss hpergeometric equations. Algebra i Analiz 7(: [] H. Movasati. On Ramanujan relations between Eisenstein series. Submitted 008. [4] H. Shiga T. Tsutsui and J. Wolfart. Triangle Fuchsian differential equations with apparent singularities. Osaka J. Math. 4(: With an appendix b Paula B. Cohen. 5