An Analogy of Bol s Result on Jacobi Forms and Siegel Modular Forms 1

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1 Journal of Mathematical Analysis and Applications 57, (00) doi:0.006/jmaa , available online at on An Analogy of Bol s Result on Jacobi Forms and Siegel Modular Forms Youngju Choie and Haesuk Kim Department of Mathematics, Pohang University of Science and Technology, Pohang, , Korea yjc@vision.postech.ac.kr, piglet@euclid.postech.ac.kr Submitted by B. C. Berndt Received December 5, 999 In this paper, the analogy of Bol s result to the several variable function case is discussed. One shows how to construct Siegel modular forms and Jacobi forms of higher degree, respectively, using Bol s result. 00 Academic Press. INTRODUCTION The classical theory of period of modular forms gives a rich structure of arithmetic of modular forms with connections such as the values of L-functions and indefinite binary quadratic forms. The concept of the period of modular forms, which is known as the Eichler Shimura map, has been developed and generalized by considering a converse of Bol s result (see, for instance, [9 ]). In this paper, the analogy of Bol s result to the several variable function case is discussed using the generalized heat operator and determinant differential operator. Applying Bol s result to Jacobi forms and Siegel modular forms, we show how to construct Siegel modular forms and Jacobi forms with different weights, respectively. This research was partially supported by BK Program and POSTECH Research fund X/0 $35.00 Copyright 00 by Academic Press All rights of reproduction in any form reserved.

2 80 choie and kim and. NOTATIONS AND DEFINITIONS Let H n be the Siegel upper half plane and Sp n be a symplectic group, Sp n = Sp n acts on H n as H n =W M n n W t = W ImW > 0 { ( )} M M n n M t 0 In JM = J J = I n 0 MW =AW + BCW + D ( ) A B for any W H n and M = Sp C D n. Let Ɣ n θ be a subgroup of the symplectic group Sp n such that { ( ) } A B Ɣ n θ = M = Sp C D n AC t BD t are even It is known that Ɣ n θ is generated by the matrices [8] ( ) U 0 tu = 0 U U GL n U =U t us = ( ) In S 0 I n even S = S t ( ) Er E r I n E I n E r E r = r ( ) Ir 0 r= 0 n 0 0 We first state the following lemma concerning the generators of the group Ɣ n θ. Lemma.. Define, for each positive integer µ, µ n, an embedding ρ µ Ɣ θ Ɣ n θ by (( )) ( ) a b In +a E ρ µ = µ µ be µ µ c d ce µ µ I n +d E µ µ where E µ µ is the n n matrix with entries E µ µ i l = δ µ i δ µ l. Then (( )) 0 ρ µ µ n 0 ( ) In S us = S = S t even 0 I n ( ) U 0 tu = 0 U U GL n U =U t generate the group Ɣ n θ.

3 an analogy of bol s result on forms 8 Proof. This follows from the fact that, for each r, r = 0 n, ( ) (( )) Er E r I n n 0 = ρ I n E r E µ r 0 µ=r+ 3. AN ANALOGY OF BOL S RESULT OF j n THE FUNCTIONS ON H n We now give a theory analogous to that of Bol for the multivariable function using heat operators. We apply Bol s result to Jacobi forms of halfintegral weight with higher degree to construct those forms with different weights. The following notations were introduced by Ziegler [4] to study Jacobi forms of higher degree. Consider the Heisenberg group n j H = λ µϰλ µ M j n ϰ M j j ϰ + µλ t =ϰ + µλ t t which is a group with the composition law λ µϰ λ µ ϰ =λ + λ µ+ µ ϰ+ ϰ + λµ t µλ t The mapping λ µϰ I n 0 0 µ t λ I j µ ϰ 0 0 I n λ t I j n j defines an embedding of H acts on H n j by multiplication from the left, into Sp n+j. Now the group Sp n λ µϰ M =λ µ M ϰ n j n j So we can define the Jacobi group G = Sp n H with the associated composition law M ζ M ζ =MM ζm ζ It can be verified that G nj acts on H n j n as a group of automorphisms. The action is given by M λ µϰ τ z =Mτ z + λτ + µcτ + D

4 8 choie and kim where M = ( A B C D). As usual, we recall the automorphy factor j n M W = θ nmw θ n W where θ n W = p Wp n eπipt, W H n. Note that the theta series θ n W satisfies the transformation law θ n MW = χ n M detcw + D / θ n W for any M = ( A B C D) Ɣn θ, where χ n M is one of the eighth roots of unity depending only on M (see []). For a holomorphic function f H n j n and any integer k, slash operators are defined as and f k+ MjMτ z =detcτ + D k j n M τ e πiσmj zcτ+d Cz t f MτzCτ + D f M jζτ z =e πiσmj λτλ t +λz t +ϰ+µλ t f τ z + λτ + µ where M = ( ) n j C D Spn ζ =λ µϰ H, half-integral positive definite M j M j j. Here, σm denotes the trace of M. We define a heat operator which acts on the space of holomorphic functions on H n j n. Definition 3. (Heat Operator). Take a positive definite matrix M j M j j. The heat operator L M j is given as ( ( ) ( ) t ( )) L M j = det 4πiM j M j τ z z where ( ) ( = + δ τ uv ( ) ( ) = z z uv τ uv ) τ =τ uv u v n H n z =z uv u j v n j n and M j is the determinant of M j, M j = M j uv, M j uv is the cofactor of the u vth entry of M j for j, and M j = when j =. Example 3.. When n = and j =, then L M is the classical heat operator L m = 8πim τ z When n = and j =, then L M = 8πiM τ i= l= M = m M il z i z l

5 When n = and j =, then L m = det an analogy of bol s result on forms 83 ( ( ) ( τ 4πim τ z z z τ τ3 z z z where τ = ( τ τ τ τ 3 ) z =z z t M = m. We have the following general lemma. )) Lemma 3.3. Let f H n j n be a smooth function, k be an integer, and M j M j j. Then, for each integer µ µ n, and ρ µ in Lemma., () One has ( LM jf k+ + Mjρ µm ) τ z = L M j( f k+ Mjρ µmτ z ) + 8πiM j ( k + n + j )( ) c cτ µµ + d µ µ L f M j k+ Mjρ µmτ z () In general, for each nonnegative integer l, one has ( L l f M j k+ +l Mjρ µm ) τ z = l ν=0 ( l ν ) ( ) Ɣαn j + l 8πiM j c l ν( L Ɣα n j + ν cτ µµ + d µ µ M j ) l νl ν M j f k+ Mjρ µmτ z with α n j = k + n + j µ µ for all M Ɣ θ where τ =τ lν. Here L is the differentialoperator M j given by the sub-determinant of L M j with the µth row and the µth column removed when j and L = when j =. ( µ µ) l ν L M j M denotes the j µ µ l νth iteration of L and the gamma function Ɣν + =ν! with M j integer ν. Proof. proof. This can be proved by induction on l, so we omit the detailed The following states the analogy of Bol s result. Theorem 3.4 (Bol s Result of the Function on H n j n ). Let f H n j n be a smooth function and n + j even. Let M M j j be

6 84 choie and kim any positive definite half integral matrix. For any nonnegative integer r and each integer µ µ n, L r+ M f r+ n+j + M ρ µm =L r+ M f r+ n+j M ρ µm Here, L M is a heat operator defined in Definition 3. and L r Mf denotes the r-composition L M L M L M f. Proof of Theorem 3.4. Letting l = r + k = r + n+j in Lemma 3.3, the theorem is obtained. To apply Bol s result to obtain Jacobi forms of higher degree we recall the definition of Jacobi forms of higher degree. Definition 3.5 (Jacobi Form of Half Integral Weight). Let k be a nonnegative integer and M M j j be a positive definite half-integral matrix. A holomorphic function f H n j n satisfying f k+ M Mτ z =f τ z M Ɣ n θ (3.) f M ζτ z =f τ z n j ζ H (3.) and having a Fourier expansion of the form f τ z = cn Re πiσnτ e πiσrz (3.3) R n j N=N t 0 N half integral Mn n which cn R = 0 only if ( N /R) /R t M 0 is called a Jacobi form of weight k + and index M. We denote the vector space of all Jacobi forms of weight k + and index M M j j on Ɣ n θ by J k M jɣ n θ. When there is no confusion on the size of M, we denote J k M Ɣ n θ instead of J k Mj Ɣ n θ. So, we can state the following result. Corollary 3.6 (Bol s Result on Jacobi Forms). Let f be a Jacobi forms of weight r + n+j and index M M j j with degree n on Ɣ n θ. Assume n + j is even. Then L r+ M f is a Jacobi form of weight r + + n+j with index M of degree n on Ɣ n θ. Proof. From Lemma., it is enough to check the first functional Eq. (3.), L r+ M f r++ n+j M ρ µmτ z =L r+ M f τ z ρ µ M µ n and it follows directly from Theorem 3.4 To check the second functional equation (3.), one notes that for any ζ =λ µϰ H f n j J k+/m jɣ n θ, L M jf M jζτ z =L M jf M jζτ z

7 an analogy of bol s result on forms 85 because, for the µ νth component of the matrix 4πiM j τ z t M j and any holomorphic function g satisfying the functional z equation we have that ( ( 4πiM j τ gτ z =e πiσmλτλt +λz t +ϰ+µλ t gτ z + λτ + µ = ( 4πiM j ) ( z ( τ ) t ( )) M j z ( ) t ( M j z z ) One can also check the Fourier expansions. g M jζ µν )) ) τ z g M jζτ z µν Remark 3.7. Theorem 3.4 and Corollary 3.6are also true for every positive integer n + j (whether n + j even or odd) as long as the slash operator is defined properly. This means that one can have an analogy of Bol s result on Jacobi forms on the other groups. In particular, the case when f is an elliptic Jacobi form of the integral weight on the group Sp =SL, i.e., when n = j =, has already been studied in [6], where the connection with the period of elliptic modular forms has been explored. 4. ANALOGY OF BOL S RESULT TO THE FUNCTIONS ON H n In this section, we state the relation between the heat operator and the determinant differential operator using the Jacobi Fourier expansion of Siegel modular forms of higher degree. An analogy of Bol s result using a determinant differential operator is obtained and used to construct the Siegel modular forms. Definition 4. (Siegel Modular Form of Half Integral Weight). We denote the vector space of Siegel modular forms of weight k + with degree n on Ɣ n θ as M k+ Ɣ nθ; let k be a nonnegative integer. A holomorphic function F H n is called a Siegel modular form of weight k + with degree n on Ɣ nθ if F satisfies F k+ MW =detcw + D k j n M W FMW = FW ( ) A B where MW =AW + BCW + D M = Ɣ C D n θ. When the degree of F is equal to, we add the holomorphic condition of F at all the cusps.

8 86 choie and kim The determinant differential operators played a role in the construction of Rankin Cohen bilinear differential operators on the Siegel modular forms of degree [4, 6]. We define the determinant differential operator n = det n n n n where uv = / τ uv, τ =τ uv u v n H n. (4.) Remark 4. [4]. Let F be in M k+/ Ɣ n+j θ. The Fourier Jacobi expansion of FW is given by FW =Fτ z τ = f M jτ ze πiσmj τ M j =M jt 0 M j half integral where τ M n n z M j n τ M j j and f M jτ z = (( T c r )) T r rt M j e πiσtτ e πiσrz where T runs all half-integral ( semi-positive definite ) matrices in M n n T /r and r M n j such that /r t M j 0. Remark 4.3. The heat operator L M j can be induced from the determinant differential operator n+j using the Fourier Jacobi expansions of Siegel modular forms of degree n + j; consider a symmetric matrix ( ) τ z t W = z τ H n+j where τ M n n, z M j n, τ M j j. We note that n = det / τ det / τ / z t / τ / z, since ( ) ( )( ) τ z t τ z t I det z τ = det n 0 z τ τ z I j ( ) τ z = det t τ z z t 0 τ = detτ detτ z t τ z Let FW be a Siegel modular form of degree n + j with a Fourier Jacobi expansion of the form FW = f M jτ ze πiσmj τ M j 0

9 an analogy of bol s result on forms 87 Since τ uv e πiσmj τ =4πim µν e πiσmj τ M j =m µν we get ( ) (e πiσm det j τ ) = ( det4πim j ) e πiσmj τ τ So, by applying the operator n+j, (( ))) τ z t n+j (F z τ = M j >0 4πi j M j det (( ) ( ) t ( ) ( )) τ z τ z f M jτ ze πiσmj τ = 4πi j n M j n L M jf M jτ ze πiσmj τ M j >0 We first state the following result. Lemma 4.4. Let F be a holomorphic function on H n and k be a nonnegative integer. Then, for each nonnegative integer l, each integer µ µ n, and ρ µ as in Lemma., one has ( l n F k+ +l ρ µ M ) ( l l ) ( ) Ɣαn + l λ c l λ W = Ɣα n + λ cw µµ + d λ=0 µ µ n l λ λ n F k+ ρ µmw µ µ where α n = k + n and W =w lν H n, M Ɣ θ. Here, n is the differentialoperator given by the sub-determinant of n with the µth row and µth column removed when n and n = when n =. Proof. details. Using induction on l, it can be proved. So we omit the The following generalizes Bol s result to the several variable functions. Theorem 4.5 (Analogy of Bol s Result to the Functions on H n ). Let F H n be a smooth function and n even. Then, for any integer µ n and 0 r, r+ n F r+ n + ρ µ M = r+ n Here, n is the differentialoperator defined in (4.). F r+ n ρ µ M

10 88 choie and kim Proof of Theorem 4.5. Substituting l = r +, k = r + n in Lemma 4.4, we obtain the result. The following corollary is an immediate consequence of Bol s result in Theorem 4.5 with Lemma.. Corollary 4.6 (Bol s Result on Siegel Modular Forms). Let F be a Siegelmodular form of weight r + n and degree n on Ɣ n θ. Then r+ n F is a Siegelmodular form of weight r + n + of degree n on Ɣ n θ. Remark 4.7. Theorem 4.5 also holds for any positive integer n, n even or odd, if the slash operator is defined properly. This means that Bol s result can be applied to the Siegel modular forms of higher genus on the other groups such as on the full Symplectic group Sp n. In particular, Bol [] originally studied the case when n =, so Sp =SL, which gave the motivation to study period theory of elliptic modular forms on the full modular group (see, for instance, [9 ]). REFERENCES. A. N. Andrianov and G. N. Maloletkin, Behavior of theta series of degree N under modular substitutions, Math. USSR Izv. 9, No. (975), G. Bol, Invarianten linearer differential gleichungen, Abh. Math. Sem. Univ. Hamburg 6, Nos. 3 4 (949), Y. Choie, Jacobi forms and the heat operator, Math. Z. 5, No. (997), Y. Choie, Multilinear operators on Siegel modular forms of genus and, J. Math. Anal. Appl. 3 (999), Y. Choie, Half integral weight Jacobi forms and periods of modular forms, Manuscripta Math. 04 (00), Y. Choie and W. Eholzer, Rankin Cohen operators for Jacobi and Siegel forms, J. Number Theory 68 (998), M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics, Vol. 55, Birkhäuser, Boston/Basel/Stuttgart, R. Endres, Über die Darstellung singulärer Modularformen halbzahligen Gewuchts durch Thetareichen, Math. Z. 93 (986), M. Knopp, Some new results in the Eicher cohomology of automorphic forms, Bull. Amer. Math. Soc. 80, No. 4 (974), M. Knopp, Rational period functions of the modular group, Duke Math. J. 45 No. (978), W. Kohnen and D. Zagier, Modular Forms with Rational Periods, Modular Functions (R. A. Rankin, Ed.), pp , Horwood, Chichester, R. Rankin, The construction of automorphic forms from the derivatives of a given form, J. Indian Math. Soc. 0 (956), D. Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. 04 (994), C. Ziegler, Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg 59 (989), 9 4.