On Ramanujan relations between Eisenstein series

Transcription

1 On Ramanujan relations between Eisenstein series Hossein Movasati Instituto de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina, 0, 460-0, Rio de Janeiro, RJ, Brazil, hossein@impa.br Abstract The Ramanujan relations between Eisenstein series can be interpreted as an ordinar diferential equation in a parameter space of a famil of elliptic curves which is inverse to the Gauss-Manin connection of the corresponding period map constructed b elliptic integrals of first and second kind. In this article we consider a slight modification of elliptic integrals b allowing non-algebraic integrants and we get in a natural wa generalizations of Ramanujan relations between Eisenstein series. Introduction In the inverse of period map of the classical two parameter Weierstrass famil of elliptic curves there appaers Eisenstein series of weight 4 and 6, and the Schwarz triangle function with triangular parameters p, q, r, where p, q, r N, is the inverse of an automorphic function for the triangle group with signature p, q, r. In all these examples the period maps of differential forms of the first kind are considered and considering periods of differential forms of the second kind, one gets differential automorphic functions which are solutions to certain ordinar differential equations (see []. Looking in this wa, it is not necessar to define (differential automorphic functions b funcational equations which the satif with respect to a Kleinian group, but as functions which are solutions to certain ordinar differential equations. To explain better this idea, let us state the main results of this paper for an example: Theorem. Consider the multi-valued function ( pm : C \{(t, t, t C 7t 4t = 0} SL(, C ( dx xdx t, where dx xdx ( = γ (7t 4t ( a ((x t t (x t t a and and are two straight paths in the x-plane connecting one root of to the other two roots. Here γ is a complex number depending onl on a, b and c, and it is taken so that the image of pm is in SL(, C.. For a z, pm is a local biholomorphism and its local inverse restricted to, 0 namel (g,a (z, g,a (z, g,a (z, where z is in some small open set U in C, satisfies

2 the sstem of ordinar differential equations: ( ṫ = t + a 9a 6 t ṫ = 4t t + a t ṫ = 6t t + 9a 6 t where dot is the derivation with respect to z. xdx. The integrals, where is a path connecting two roots of, are constant along the solutions of (.. g k,a s with repsect to the group (4 Γ := M SL(, C, M := i e πia 0 e πia := i e πia e πia 0 e πia, have the following automorphic properties: for ever A = Γ and z U c d such that cz + d 0 there exists an analtic continuation of g k,a, k =,, along a path which connects z to Az such that (5 (cz + d k g k,a (Az = g k,a (z, k =,, (6 (cz + d g,a (Az = g,a (z + c(cz + d. One can show that (7 g k, ( = a k + ( k 4k B k n σ k (ne πizn, k =,,, z H, is the Eisenstein series of weight k, where H is the upper half plane, B k is the k-th Bernoulli number (B = 6, B = 0, B = 4,..., σ i(n := d n di and (8 (a, a, a = ( πi, (πi, ( πi (see []. In the case a = the sstem of ordinar differential equations ( is known as the Ramanujan relations between g k,a, k =,, because he had observed that in this case the series (7 satisf the differential equation ( (see for instance [8]. I do not know an explicit expressions like (7 for an arbitrar a C. Throughout the text we will consider the famil (9 which is a generalization of ( and it has the advantage that it contains full Gauss hpergeometric functions. The text is organized in the following wa: In we consider a more general famil of transcendent curves. In and 4 we fix up the paths of integration and calculate the monodromies. In 5 we calculate the derivation of the period map. The calculation is similar to the calculation of Gauss-Manin connections in the algebraic context. In 6 we calculate the determinant of the period map and according to this calculation in 7 we redefine the period map. In 8 we take the inverse of the period map and obtain Ramanujan tpe relations. 9 is devoted the action of an algebraic group. Finall in 0 we discuss the automorphic properties of the functions which appear in the inverse of the period map. I would like to thank IMPA in Rio de Janeiro and MPIM in Bonn for their lovel research ambient.,

3 Families of transcendent curves For a, b, c C fixed, we consider the following famil of transcendent curves: (9 E t,a,b,c = E t : = f(x, f(x := t 0 (x t a (x t b (x t c. Here t = (t 0, t, t, t C 4 is a parameter. The discriminant of E t is defined to be = (t := t 0 (t t (t t (t t and we will work with regular parameters, i.e. t T := {t C 4 (t 0}. The parameter t 0 is just for simplifing the calculations related to the Gauss-Manin connection of the famil (see 5. If a, b and c are rational numbers then E t s are algebraic curves. In this case one can use algebro-geometric methods in order to stud the periods of E t, see for instance [4]. In general, E t is a solution of the following logarithmic differential equation d = adx x t + bdx x t + cdx x t. In order to prove Theorem we also consider the famil (0 Ẽ t : = f(x, f(x = t 0 ((x t t (x t t a. In the case a = b = c there is a canonical map from the parameter space of the first curve to the parameter space of the second curve: t 0 = t 0, t = t + t + t, t = ( t t ( t t + ( t t ( t t + ( t t ( t t, t = ( t t ( t t ( t t. For simplicit we will also use t instead of t; being clear parameters of which famil we are talking about. Paths of integration and Pochhammer ccles We distinguish three, not necessaril closed, paths in E t. In the x-plane let 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 t which are mapped to i under the projection on x. We choose one of them and call it i. For the case in which Re(a, Re(b, Re(c < 0 the paths i s and i s are depicted in Figure. We can make our choices so that + + is a limit of a closed and homotopic-to-zero path in E t. For instance, we can take the path i s in such a wa that the triangle formed b them has almost zero area. Now, we have the integral ( p(xdx = p(xdx f(x, p C[x], where 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 [7].

4 plane =f(x t ~ t ~ ~ t x plane Figure : Paths of integration Another wa to stud the integrals ( 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 is the commutator α = [γ, γ ] = γ γ γ γ, where γ is a loop along starting and ending at some point in the midle of which encircles t once anticlockwise, 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 (see [7], Proposition..7. Note that in order to have d( p(x = 0, p C[x], i =,, i f(x α p(xdx f(x dx. we have to assume that Re(a, Re(b, Re(c < 0. But this is not necessar if we work with Pochhammer ccles. 4 The period map and the monodrom group For a fixed a T, the period map is given b: ( ( pm : (T, a GL(, C, t dx dx xdx xdx, 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 aa. We denote b P the union of images of the extensions of pm and call it the period domain. In order to stud the analtic extensions of pm we have to calculate the monodrom group. In what follows we use the following convention: Two paths in E t are equal if the integration of an differential form p(xdx, p C[x] over them is equal. For instance, using this convention we have ( + + = 0. 4

5 Let A = e πia, B = e πib, C = e πic. We fix t and t and let t turn around t anti clockwise. We obtain three new paths h (, h ( and h ( in E t such that h ( + h ( + h ( = 0 (this follows from (. Note that in the x-plane (resp. in E t the triangle formed b h ( i s (resp. h ( i s does not intersect itself. We have h ( = + (A AB, h ( = A = + ( A, h ( = AB (see Figure, A. We call h the monodrom around the hperplane t = t. These formulas are compatible with the Picard-Lefschetz formula in the case a = b = c =. In a similar wa and h ( = + B BC, h ( = B, h ( = BC h ( = + C CA, h ( = C, h ( = CA. Therefore, the monodromies with respect to the basis (, are represented as ( A A BC 0 C CA M = A(B A(B + = B =. 0 CA Note that and that for n N M M M = ABC 0, 0 ABC h n ( = +(A AB (ABn AB, h n ( = +( A (ABn AB, h n ( = (AB n. The monodrom group Γ is defined to be the subgroup of GL(, C generated b M i, i =,,. For a = b = c = we have 0 M = = = 0 and it is eas to see that Γ = Γ( := {A SL(, Z A Id}. We discuss the monodromies for the famil (0. In this famil the monodromies change the place of the roots of f. Therefore, the monodrom h corresponding to the replacement of t and t is given b h ( = = +, h ( = + A, h ( = A. (see Figure B. The other monodromies are h ( =, h ( = + A, h( = A. h ( =, h ( = + A, h( = A. Therefore, in the basis (,, the monodromies has the form 0 A 0 M = A A = = 5 A. 0 A

6 t t t t t t t t t A B Figure : Monodrom Note that For a = we have M = 0 = M M M = M 0 =. 0 and so Γ = SL(, Z. Remark. In general it is hard to decide for which parameters a, b, c the group Γ is Kleinian, i.e it acts discontinuousl in some open subset of C { }. There is a necessar condition for such groups called Jorgensen s inequalit (see [] but it is not sufficient. For ν 0 := a c = p, ν := b c = q, ν := a b = r, where p, q, r are positive integers, the group Γ is the triangular group of tpe p, q, r and it is Kleinian (see [, 9, 4]. Despite the fact that Γ\P ma not have an reasonable structure, the global period map pm : T Γ\P is well-defined. 5 A kind of Gauss-Manin connection The Gauss-Manin connection is the art of derivation of differential forms on families of algebraic varieties and then simplifing the result. Despite the fact that the varieties considered in this article are not algebraic, the process of derivation and simplification is similar to the algebraic case (see for instance [0]. In what follows, derivation with respect to x is denoted b. First of all we have to simplif the integral (. More precisel we want to reduce it to the integrals with p =, x. Let R = C(t and g = (x t (x t (x t. Proposition. For all p R[x], there is p R[x], deg( p such that pdx = pdx, where is a path which connects two points of {t, t, t } and does not cross it elsewhere. I would like to thank Katsuhiko Matsuzaki who informed me about the mentioned fact. 6

7 Proof. For n > modulo exact forms we have 0 = d( xn g = ( x n g f f f + (xn g dx f. Note that g f f is a polnomial in x. We set p n = a n x n + r n (x, a n C, deg(r n n the polnomial in the parenthesis. We have a n 0 and so modulo exact forms we have: x n dx f = a n r n dx f. B various applications of the above equalit in pdx we finall get the desired equalit. Let us now differentiate the integrals: Proposition. Let t be one of the parameters t i, i = 0,,,. We have pdx pdx = t f t f, p C[x], where pdx := t f ((a f t g f t g p + a f Proof. We can find two polnomials a, a R[x] such that We have g f f a + ga =. f p + p t dx f. pdx t f = = ( f t p + p f t dx f ( f t gp f g + p t dx f = ( p g + p f t dx f, t p = g p f = ( (g f f a + ga p + p g t dx f = ( df f a p + a p dx f + p dx t f = ( f d(a p + a p dx f + p t = ((a p + a p + p t dx f. dx f 7

8 For the implementation of the algorithms of this section in Singular [] see the author s web page. For the famil (9 we have used these algorithms and we have obtained: (4 t ω = where at + (a + c t + (a + b t a b c + ω (t t (t t at t + (b t t + (c t t ( a b c + t ω = ( dx xdx The derivations with respect to t (resp t is obtained b permutation of t with t and a with b (resp. t with t and a with c. It is also eas to check b hand that t 0 ω = t ω We will simpl denote b, i = 0,,, the matrix A in ω = Aω. t i t i For the famil (0 we use g = (x t t (x t t and = t 0 (4t 7t and we have t = t = ( (4t 7t (4t 7t t 0 = t 0 0 0, = t 0 0, 0 7at t 6at + 8t t + t 7at 8t 7at t + 9at t + 8t t t t t t 7at t 6at 8t t + 4t ( 8at t + 7at t t 9t 8at t 6at t t + 9t t + t 8at + t. 8at t + 7at + t t 8t, 6 Determinant of the period matrix From Proposition it follows that the period map satisfies the differential equation d(pm = pma tr, where A = i=0 ( t i dt i. This and (4 impl that det := det(pm satisfies det t = (t t (t t ((a + c t + (a + b t + ( a b c + t det. Solving this differential equation one concludes that det is of the form C(t t a c (t t a b, where C does not depend on t. Repeating the same argument for t 0, t, t we conclude that (5 det(pm = γ t 0 (t t a c (t t a b (t t b c, where γ is a constant depending onl on a, b and c. For the famil (0 in a similar wa we get det(pm = γ t 0 (7t 4t a. 8

9 7 Redefining the period map and the monodrom group We have calculated the determinant of the period map in (5. It depends on t, t, t except for the case a = b = c =. In order that the determinant of the period map to be equal to t 0 and the monodrom group to be a subgroup of SL(, C, we have to multipl ( b p := γ (t t ( a c (t t ( a b (t t ( b c. In other words, we have to redefine f(x := γ t 0 (t t ( a c (t t ( a b (t t ( b c (x t a (x t b (x t c for the famil (9. We have to recalculate the Gauss-Manin connection. B Leibniz rule we have (pω = (dp ω + p A ω = ( dp p I + A (pω and dp p = (a + b dt dt t t + = ( (a + b t t + (a + c t t dt + After redefining the period map the monodrom matrices are changed as follows: M = ( A A AB A(B A(B + = BC 0 BC B = C CA CA 0 CA Note that A = e πia, are well-defined and Γ := M = M SL(, C. For the famil (0 we redefine f(x = γ t 0 (7t 4t ( a ((x t t (x t t a, t = (t 0, t, t, t C 4 which is the one in ( with t 0 =. For p = (7t 4t ( a we have dp p = (a 54t dt t dt 7t 4t The new monodrom group is (4. For both families we conclude that det(pm = t 0. Remark. A subgroup A of SL(, R is called arithmetic if A is commensurable with SL(, Z, i.e. A SL(, Z has a finite index in both SL(, Z and A. It is a natural question to classif all the cases such that the monodrom group is conjugated with an arithmetic group. For the case a = b = c = 6 the monodrom group is conjugated with SL(, Z (see []. 8 The inverse of the period map First we note that the period map is a local biholomorphism. We consider pm as a map sending the point (t 0, t, t, t to (x, x, x, x 4. Its derivative at t is a 4 4 matrix whose i-th column constitutes of the first and second row of x( t i tr. For s := a + b + c 0, this is an invertible matrix. More precisel, we have (df x = (dpm t = det(x 9

10 0 t 0 x 4 t 0 x B (at t x + at t x at x 4 at t x + bt x bt x 4 + ct x ct x 4 t x t t x t t x + t x 4 + t t x /s ( t x + x (at x at x 4 + bt t x bt t x + bt t x bt x 4 + ct x ct x 4 t t x + t t x t x t t x + t x 4 /s ( t x + x 4 (at x at x 4 + bt x bt x 4 ct t x + ct t x + ct t x ct x 4 + t t x t t x t t x t x + t x 4 /s ( t x + x 4 t 0 x t 0 x ( at t x at t x + at x + at t x bt x + bt x ct x + ct x + t x + t t x + t t x t x t t x /s (t x x ( at x + at x bt t x + bt t x bt t x + bt x ct x + ct x + t t x t t x + t x C + t t x t x /s (t x x A ( at x + at x bt x + bt x + ct t x ct t x ct t x + ct x t t x + t t x + t t x + t x t x /s (t x x and F = (F 0, F, F, F : (P, x 0 (T, a is the local inverse of pm, where x 0 = pm(a. From det(pm = ( t 0 it follows that F 0 (x = det(x z0. Let us take a in such a wa that x 0 is of the form. Let g 0 i (z z be the restriction of F i to, where z is in a neighborhood of z 0 0 in C. Considering the equations related to the entries (i,, i =,, 4, we conclude that (g (z, g (z, g (z satisfies the ordinar differential equation: (6 a ṫ = a+b+c (t t + t t t t + ṫ = b a+b+c (t t + t t t t + ṫ = c a+b+c (t t + t t t t + b+c a+b+c t a+c a+b+c t a+b a+b+c t In a similar wa for the famil (0, we get ( and so the first part of Theorem is proved. Let Ra be the vector field in C 4 corresponding to (6 together with ṫ 0 = 0. It is a mere calculation to see that 0 Ra = This means that d(pm(ra = and so xdx 0 is constant along the leaves of F(Ra. The similar argument work for the famil (0 and so the second part of Theorem is proved.. 9 Action of an algebraic group The algebraic group { } k k (7 G 0 = k 0 k C, k, k C acts on the period domain P GL(, C from the right b the usual multiplication of matrices. It acts also in C 4 as follows: t g := (t 0 (k k, t k k + k k, t k k + k k, t k k + k k (8 t = (t 0, t, t, t C 4 k k, g = G 0 k 0. The relation between these two actions of G 0 is given b: When the paper was finished, I found that such a differential equation was alread discovered b G. Halphen [6, 5, 4] in his stud of hper-geometric functions. However, the geometric interpretation and the automorphic properties of its solutions are new in this paper. 0

11 Proposition. We have (9 pm(t g = pm(t g, t T, g (G 0, I where (X, x means a small neighborhood of x in X and I is the identit matrix. If t s, s [0, ] is a path in T and g s, s [0, ] is a path in G 0 which connects I to g G 0, b analtic continuation of the equalit (9, it makes sense to sa that (9 is true for an arbitrar g G 0. Proof. Let Then α : C C, (x, (k k x k k, k k. k k α ( f(x = (8πit 0 k k (t t ( b c (k k x k k t a = (8πit 0 k a b c k a+b+c (k k ( (a+b+c (k k t k k t ( b c (x (k k t +k k a = (8πit 0 (k k (k k t +k k (k k t +k k ( b c (x (k k t +k k a This implies that α induces an isomorphism Now ( α ω = where ω = ( dx, xdx tr, and so which proves (9. α : E t g E t. k 0 k k k k ω = pm(t = pm(t g.g In a similar wa for the famil (0 we have the action k 0 ω, k k t g := (t 0 k k, t k k + k k, t k k, t k 4 k (0 t = (t 0, t, t, t C 4 k k, g = G 0 k 0 with the propert (9. Remark. The rational map α : C C, (x, (x, (x t [a] (x t [b] (x t [c] sends E t,a,b,c biholomorphicall to E t, a, b, c. We use Proposition and write α ω tr = ω tr C, C Mat(, Q[t]. The period map associated to E t, a, b, c is the multiplication of the period map associated to E t,a,b,c with C. For this reason it is sometimes practical to assume that 0 Re(a, Re(b, Re(c <.

12 0 Automorphic properties of g i s We continue the notation introduced in 8. We denote b F = (F 0, F, F, F : (P, x 0 (T, a the local inverse of the period map. Taking F of (9 we conclude that ( F (xg = F (x g, g (G 0, I. We get F 0 (xg = F 0 (xk k, ( F i (xg = F (xk k + k k, i =,, The first equalit ( also follows from F 0 (x = det(x. For an A = Γ there is a path γ π c d (T, a such that if pm : (T, a P is the analtic continuation of pm along γ then pm(t = Apm(t, t (T, a. This implies that the analtic continuation of F along the path = pm(γ, which connects pm(a to Apm(a satisfies ( F (x = F (Ax, x (P, x 0. Using the Schwarz function D(t = dx f dx f we define the path σ = D(γ. If cz 0 + d 0 then Az 0 is well-defined and the path σ connects z 0 to Az 0 in C. We claim that there is an analtic continuations of g i s along σ such that (cz + d g i (Az = g i (z + c(cz + d, i =,,, A = Γ, z (C, z c d 0. We have z (, g, g, g = F 0 ( ( a b z = F c d 0 ( Az cz + d c = F 0 0 (cz + d det(a ( ( Az cz + d c = F 0 0 (cz + d = (, (cz + d g (Az c(cz + d,

13 The fourth equalit makes sense in the following wa: Let D(γs x s := P, τ 0 s := x s pm(γ s G 0, s [0, ]. cz + d c τ is a path in G 0 which connects I to 0 (cz + d. For s near enough to 0 we have F (x s τ s = F (x s τ s and so b analtic continuation we have it for s =. In a similar wa we prove the third part of Theorem. Note that for the famil (0, F and F satisf: F (xg = F (xk k, F (xg = F (xk 4 k, x L, g G 0. References [] Tom M. Apostol. Modular functions and Dirichlet series in number theor, volume 4 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 990. [] A. Beardon, The geometr of discrete groups. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 995. [] G.-M. Greuel, G. Pfister, and H. Schönemann. Singular.0. A Computer Algebra Sstem for Polnomial Computations, Centre for Computer Algebra, Universit of Kaiserslautern, [4] G. Halphen. Sur certains sstéme d équations différetielles. C. R. Acad. Sci Paris, 9: , 88. [5] G. Halphen. Sur des fonctions qui proviennent de l équation de gauss. C. R. Acad. Sci Paris, 9: , 88. [6] G. Halphen. Sur une sstéme d équations différetielles. C. R. Acad. Sci Paris, 9:0 0, 88. [7] Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, and Masaaki Yoshida. From Gauss to Painlevé. Aspects of Mathematics, E6. Friedr. Vieweg & Sohn, Braunschweig, 99. A modern theor of special functions. [8] Serge Lang. Introduction to modular forms, volume of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 995. With appendixes b D. Zagier and Walter Feit, Corrected reprint of the 976 original. [9] J. Lehner Discontinuous groups and automorphic functions. Mathematical Surves, No. VIII American Mathematical Societ, Providence, R.I [0] H. Movasati. Calculation of mixed Hodge structures, Gauss-Manin connections and Picard-Fuchs equations, Real and Complex Singularities, Trends in Mathematics, 47-6, 006, Birkhauser. [] H. Movasati. On Differential modular forms and some analtic relations between Eisenstein series, to appear in Ramanujan Journal.

14 [] H. Movasati, S. Reiter, Hpergeometric series and Hodge ccles of four dimensional cubic hpersurfaces, Int. Jou. of Number Theor, Vol., No. 6, 006. [] Y.V. Nesterenko and P. Philippon. Introduction to algebraic independence theor, volume 75 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 00. With contributions from F. Amoroso, D. Bertrand, W. D. Brownawell, G. Diaz. Laurent, Yuri V. Nesterenko, K. Nishioka, Patrice Philippon, G. Rémond, D. Ro and M. Waldschmidt, Edited b Nesterenko and Philippon. [4] Hironori Shiga and Jürgen Wolfart. Algebraic values of triangle schwarz functions, Proceedings of CIMPA Summer School, Istanbul