Linear Ordinary Differential Equations Satisfied by Modular Forms

Transcription

1 MM Research Preprints, 0 7 KLMM, AMSS, Academia Sinica Vol. 5, December 006 Linear Ordinary Differential Equations Satisfied by Modular Forms Xiaolong Ji Department of Foundation Courses Jiyuan Vocational Technical College Henan , P.R. China jixiaolong08@sohu.com Yujie Ma Key Laboratory of Mathematics Mechanization Chinese Academy of Sciences Beijing 00080, P.R. China yjma@mmrc.iss.ac.cn Abstract. We present an efficient algorithm to calculate the linear homogeneous ordinary differential equations satisfied by modular forms.. Introduction Let Γ be a discrete subgroup of SL (R) commensurable with SL (Z). In the theory of modular forms, P. F. Stiller s result asserts that if t(τ) is a non-constant meromorphic modular function of weight zero F (τ) a meromorphic modular form of weight with respect to a multiplier system of Γ, then F, τf,, τ F, as functions of t, are linearly independent solutions of a ( + )-st order linear homogeneous ordinary differential equation with algebraic functions of t as coefficients. In [5], the author presents a new proof of this result by using the theory of modular functions presents the following theorems. Theorem. Let Γ be a discrete subgroup of SL (R) commensurable with SL (Z). Suppose that t = t(q) is a non-constant (meromorphic) modular function invariant under Γ, F (t) = F (t(q)) is a (meromorphic) modular form of weight on the group Γ with respect to a multiplier system χ. Then the functions F (t), τf (t),, τ F (t) are linearly independent solutions of a ( + ) st order linear differential equation of the form () Dt + F + r (t)dt F + r (t)dt F + + r 0 (t)f = 0, where D t = t d dt r m(t) are algebraic functions of t. Theorem. Let t F be given in Theorem., suppose tha satisfies the (+)- st order linear differential equation () Dt + F + r (t)dt F + r (t)dt F + + r 0 (t)f = 0,

2 Linear Ordinary Differential Equations where r m (t) are algebraic functions of t. Then the function Q(t) in the Schwarzian differential equation ( ) dt Q(t) + t, τ = 0 dτ satisfied by t is equal to Q(t) = + c r + c tr + c 3r 3 4t, where c m are absolute constants depending only on t, τ denote the Schwarzian derivative of t w.r.t. τ. In this paper, we present an efficient algorithm to calculate the linear homogeneous ordinary differential equations satisfied by a given modular forms. This paper is organized as follows. In Section, we introduce the modular form Schwarzian differential equation. In Section 3, we present the explicit formula of r (t) r (t) by p (t) p (t), hence we get the explicit formula of Q(t) in terms of r (t), r (t) their derivatives. In Section 4, we present efficient algorithm to calculate the linear homogeneous ordinary differential equations satisfied by a given modular forms.. Preliminary Le F be two linearly independent solutions of the following linear differential equation F + p (t)f (t) + p (t)f = 0. Define a variable τ = F (t)/f (t). It is well nown that F = W dt dτ, where W = F F F F is the Wronsian, Abel s identity states that t W = c exp p (u)du for some constant c depending only on the choices of F F. equalities, we can deduce that F = F ( ẗ ṫ p ), F = F p ẗ p 4 ṫ 3ẗ ṫ 4 p + ẗ ṫ 3, From the above two where f f denote the derivatives of f with respect to t τ, respectively. Substituting these two identities to the original differential equation one sees that the function t(τ) satisfies a non-linear differential equation ( ) dt Q(t) + t, τ = 0 dτ

3 Yujie Ma with t, τ = ẗ ṫ 3 ( ) ẗ ṫ, Q(t) = 4p tp p 4t. This differential equation is nown as the Schwarzian differential equation in the literature, t, τ is called the Schwarzian derivative of t = t(τ). In [5], the author stated that give a ( + )-st differential equation () satisfied by a modular form F of weight it is easy to see from Theorem. that the algebraic functions p p can be recovered from the first two coefficients r r of the differential equation. On the other h, the coefficients r,, r 0 are uniquely determined by p p. Thus, the functions r m, m = 0,,, can be expressed as polynomials of t, r, r their derivatives. And the author presents such relations for the case = explicitly. The author states that for general it is not difficult to show that r = ( + )p / r = d p + d tp + d 3p for certain numerical constants. Moreover, the author asserts that the function Q(t) in the associated Schwarzian differential equation can be written as the following form Q(t) = + c r + c tr + c 3r 4t, where c m s are absolute constants depending only on. 3. Differential Equations satisfied by modular forms In this section, we will determined the coefficients in Theorem. Theorem. explicitly. Lemma 3. s n,n = p n(n ) n ( j ). Proof: By Dt n F = D t (Dt n F ), we have the following recursive formulae on s n i,n i for i = 0,, n. j= s n,n = ( n n )s n,n p (n ) ( j ) j= s n,n = ( n 3 n 3 )s n,n 3 p (n ) ( j ) Since s, = p, by elementary calculation mathematical induction, we have s n,n = p n(n ) n ( j ). j= j=

4 Linear Ordinary Differential Equations 3 Lemma 3. with s n,n = p f (n) tp g (n) p h (n) n 3 f (n) = ( j n [ ] (m )(m ) ) j= m= n 3 g (n) = ( j n [ ] (m )(m ) ) j= m= n 3 h (n) = ( j n [ ) (m )( m ] ) j= m= Proof: By Dt n F = D t (Dt n F ), we have the following recursive formulae on s n i,n i for i = 0,, n. s n,n = ( n 3 )s n,n 3 p (n ) n j= ( j ) (n )p s n,n + D t (s n,n ) s n,n 3 = ( n 4 )s n,n 4 p (n ) n 3 j= ( j ) (n 3)p s n,n 3 + D t (s n,n 3 ) Since s,0 = p, by elementary calculation mathematical induction, we then prove the lemma. Theorem 3.3 Le be a modular form satisfying the differential equation (). Then r = ( + )p / with Corollary 3.4 r = d p + d tp + d 3 p d = d = 3 6 = d 3 = p = ( + )( + ) =, 6 ( )( + ), 6 ( )( + )(3 + ) =. 4 r ( + ) p = 6( + )r + ( + 3 )r + t( )r ( + ). ( + )

5 4 Yujie Ma Proposition 3.5 The function Q(t) appeared in the associated Schwarzian differential equation can be write as the following form where Q(t) = + c r + c tr + c 3r 4t c = 4 + 4, c =, c 3 =. Note that the description of Q(t) in [5] is problematic. Proposition 3.6 The function Q(t) appeared in the associated Schwarzian differential equation can be write as the following form Q(t) = tr 5 + tr 4 r + r 4 r 4t (+) (+) = tr +5 tr ++4r r +4r 4t (+) (+) = ( ) t(+)r r +4(+)r 4t (+) (+) = +c r +c tr +c 3r 4t = 3 4(+)(+) t (+)(+) t 4(+)(+) t 3r 3r t(+) + (+) t(+) (+) (+)(+) t 3r + 6r + 6r. (+)(+) t (+)(+) t (+)(+) t where 4 c = c = c 3 = (+)(+), (+)(+),. (+)(+) 4. Algorithm Following the proof of Theorem. of [5], we have the following theorem on the algorithm for the explicit formula of the linear homogeneous ordinary differential equations satisfied by a given modular form. Theorem 4. Let Γ be a discrete subgroup of SL(, R) commensurable with SL(, Z). Suppose that t = t(q) is a non-constant (meromorphic) modular function invariant under Γ. Then the functions F (t), τf (t),, τ F (t) are linearly independent solutions of a (+) st order linear differential equation of the form () Dt + F + r (t)dt F + r (t)dt F + + r 0 (t)f = 0, where r m (t) are algebraic functions of t. The equation can be explicitly obtained by a polynomial time algorithm.

6 Linear Ordinary Differential Equations 5 Proof: First define D t = t d dt, D q = q d dq with q = exp π τ for Imτ > 0. Set G = D qt, G = D qf t F. We have D = t D qf D q t = tf G = F G tg G G D t = t D q G t G D q G G G D q t G = D qg D q t G G G 3 D q G. By Lemma of [5], the functions D q G G G / D q G G / are meromorphic modular forms of weight 4, so is G. Therefore, the following functions are rational functions of t. Thus, we have p (t) = D qg G G / G p (t) = D qg G / G G D t = G G G G p p. G We then can compute the higher order derivatives of F inductively, Dt F = D t (D ) = D t (F G ) = F ( G )G G G p p G G 3 G 3 G G = D F + p G G + p Dt 3 F = D t (Dt F ) = F G +F ( ) G = ( )( ) = F G = D F + p D F + p ( ) G G p G G p G D t G G tp G G G p D t G tp ( )( ) G3 + ( 3 G 3 3)p G G +[(/ 3)p tp + p ] G G + p p tp = D 3 ( )( ) F ( 3 3)p G [(/ 3)p G tp + p ] G G p p + tp = D 3 ( )( ) F ( 3 3)p D F + p DtF F + p [(/ 3)p tp + p ] DtF F p p + tp D 3 F + 3p D F [(/ 3)p tp p ] DtF F + p p + tp (7)

7 6 Yujie Ma In general, the n-th derivative of F taes the following form Dt n G n n G n G n F = F G n ( j/) + s n,n G n + s n,n G n + + s n,0, j= where s n,j are polynomials of t, p, p their derivatives which can be explicitly determined by chain rule of derivatives inductively. When n = +, the term of D + t Dt + F = F s +, G G F involving G+ G + vanishes, we have G + s +, G + + s +,0, Therefore, for n, the Gn G n s are linear sum of lower order derivatives of F with coefficients as polynomials of t, p, p their derivatives, henceforth, Dt + is equal to a linear sum of lower order derivatives of F with rational functions of t as coefficients. Since the rational functions p (t) p (t) can be calculated explicitly by Magma, we have the following algorithm. Algorithm: Input: F (t) = F (t(q)) is a (meromorphic) modular form of weight on the group Γ with respect to a multiplier system χ. Output: The coefficients of the ( + ) order linear homogeneous ordinary differential equation satisfied by F.. Compute G = Dqt t G = DqF F.. Compute p (t) = DqG G G / G 3. Compute D i tf for i from to +. p (t) = DqG G /. G 4. Compute ( G G ) i for i from to. Substitute the ( G F to obtain the linear ordinary differential equations. D + t 5. Transform the resulted equation by taing D t = t d dt. Acnowledgments G ) i s into the equation involved This wor is partially supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the National Key Basic Research Project of China (No. 004CB38000), the Sino Germany Center Project (GZ30) the National Natural Science Foundation of China (No ).

8 Linear Ordinary Differential Equations 7 References [] J. M. Borwein, P. B. Borwein, Pi the AGM. A study in analytic number theory computational complexity, Reprint of the 987 original, A Wiley-Interscience Publication, John Wiley & Sons Inc., New Yor, 998. [] B. H. Lian S.-T. Yau, Mirror Maps, Mudular Relations Hypergeometric Series I. Preprint, [3] B. H. Lian S.-T. Yau, Arithmetic properties of mirror maps quantum coupling. Comm. Math. Phsy. 76 (996), [4] P. Stiller, Special values of Dirichlet series, nomodromy, the periods of automorphic forms. Mem. Am. Math. Soc. 49 (99), iv+6 (984) [5] Y. Yang, On differential equations satisfied by modular forms, Math. Z. 46 (004), 9.