MULTILINEAR OPERATORS ON SIEGEL MODULAR FORMS OF GENUS 1 AND 2

MULTILINEAR OPERATORS ON SIEGEL MODULAR FORMS OF GENUS 1 AND 2 YOUNGJU CHOIE 1. Introduction Classically, there are many interesting connections between differential operators and the theory of elliptic modular forms and many interesting results have been explored. In particular, it has been known for some time how to obtain an elliptic modular form from the derivatives of N elliptic modular forms, which has already been studied in detail by R. Rankin in [9] and [10]. When N = 2, as a special case of Rankin s result in [10], H. Cohen constructed certain covariant bilinear operators which he used to obtain modular forms with interesting Fourier coefficients[4]. Later, these covariant bilinear operators were called Rankin-Cohen operators by D. Zagier who studied their algebraic relations[12, 13] as well as connections with formal pseudodifferential operators[5]. Furthermore, Rankin-Cohen operators are shown to appear as the various terms in the (convergent) expansion of the composition of two symbols in a certain symbolic calculus associated with SL(2, R)[11]. Recently, the Rankin-Cohen type bracket operator on Jacobi forms and Siegel modular forms has been studied using the heat operator and differential operator, respectively. In fact, it was shown how to construct, explicitly, Jacobi forms and Siegel modular forms of genus 2 with the Rankin-Cohen type bracket operators involving the heat operator and the determinant-differential operators, respectively[1]. In this paper, we give an explicit construction of multilinear covariant differential operators on the space of Siegel modular forms of genus 1 and 2. This shows how to obtain a Siegel modular form from the derivatives of N Siegel modular forms as an analogous result of that given by R.Rankin[10] even if the method of obtaining such forms are very different from that by R.Rankin. We generalize the Rankin-Cohen type bilinear covariant differential operators, which were studied in [3], on the spaces of Jacobi forms and Siegel forms. However, it seems not clear how to use of generating functions to produce multilinear differential operators on the space of Siegel modular forms of genus 2 from the proofs given in [3]. We overcome this difficulty by using the exponential differential operators. We thank the referee for the many helpful comments which have improved our exposition. 2. Definitions and Notations Let us recall the basic definitions and notations[8]. Let H n be the Siegel upper half plane H n and SP 2n (R) be a symplectic group,i.e., H n = {Z M n n (C) Z t = Z, Im(Z) > 0}, 1991 Mathematics Subject Classification. Primary 11F12, 11F66, 11M06; Secondary 14K05. This research was partially supported by BSRI 97-1431, KOSEF 971-0101-003-2 and RCAA98. 1

2 YOUNGJU CHOIE and SP 2n (R) = {M M 2n 2n (R) M t JM = J}, with J = ( 0 n ) I n 0 n. Then SP 2n (R) acts on H n as (M, Z) M Z = (AZ + B)(CZ + D) 1, M = ( A B C D ) SP 2n(R). Define, for a function F : H n C and integers k, the slash operator as (F k M)(Z) = Det(CZ + D) k F (M Z), M = ( A B C D ) SP 2n(R). Now, let Γ n be the Siegel modular group SP 2n (Z). We define a Siegel modular form of weight k with genus n on Γ n. Definition 2.1. A holomorphic function F : H n C is called a Siegel modular form of weight k with genus n on Γ n if F satisfies (F k M)(Z) = F (Z), M Γ n. In the case n=1, we also require that F is holomorphic in. We denote the vector space of Siegel modular forms of weight k with genus n on Γ n as M k (Γ n ). The main results involve the determinant-differential operator which has already been studied in the context of bilinear differential operators[3]. Definition 2.2. The determinant-differential operator D n is given as D n = Det From now on, we denote 2.. z 11 z 12 z 1n 2.. z 12 z 22 z.... 2n.. 2 z 1n znn z i,j as i,j. So D n = Det I n, with Z = (z i,j ) 1 i,j n H n. ( 211 12.. 1n 12 2 22.. 2n.... 1n.. 2 nn ). 3. Construction of Multilinear operators on the spaces of Siegel forms In this section, we study multilinear operators and give an explicit construction of Siegel modular forms with genus 1 and 2. This shows how to obtain a Siegel modular form from the derivatives of N Siegel modular forms as an analogous result of that given by R.Rankin[10] even if the method of obtaining such forms are very different from that by R.Rankin. We generalize the Rankin-Cohen type bilinear covariant differential operators, which were studied in [3], on the spaces of Jacobi forms and Siegel forms. However, it seems not clear how to use of generating functions to produce multilinear differential operators on the space of Siegel modular forms of genus 2 from the proof given in [3]. We overcome this difficulty by using the exponential differential operators. We point out that the most theorems stated in this section can be applied to Siegel modular forms of genus n, but because of the Leibnitz rule of the determinant operator D n, our conclusion is valid only for the case when genus n = 1 or 2. The detailed explanation is given in Remark3.8. We begin with the following two lemmas.

MULTILINEAR OPERATORS ON SIEGEL MODULAR FORMS OF GENUS 1 AND 2 3 Lemma 3.1. [6] Let Γ n be the Siegel modular group. Define, for each positive integer 1 µ n, an embedding ρ µ : Γ 1 Γ n by ( ) ( ) a b In + (a 1)E ρ µ ( ) = µ,µ be µ,µ c d ce µ,µ I n + (d 1)E µ,µ where E µ,µ is the n n matrix with entries (E µ,µ ) j,k = δ µ,j δ µ,k. Then ( ) ( ) 0 1 In T ρ µ ( ), (1 µ n, T = T 1 0 0 I t M n,n (Z)). n generates the Siegel modular group Γ n. (Proof in [6]) First, note that ρ µ indeed defines an embedding into Γ n since (I n + (a 1)E µ,µ )(be µ,µ ) t = (be µ,µ )(I n + (a 1)E µ,µ ) t, (ce µ,µ )(I n + (d 1)E µ,µ ) t = (I n + (d 1)E µ,µ )(ce µ,µ ) t, (I n + (a 1)E µ,µ )(I n + (d 1)E µ,µ ) t = I n + (be µ,µ )(ce µ,µ ) t. Next, it is well known that Γ n is generated by the elements ( ) ( ) 0 In In T S =, T = (T = T I n 0 0 I t M n,n (Z)). n The lemma follows now from the simple observation that S = n ρ µ (S) µ=1 where S = ( 1 0 1 0 ) (the product actually makes sense since the ρ µ(s) s commute). Lemma 3.2. Let F be a holomorphic function on H n. Then, for each positive integer 1 µ n and ρ µ as in Lemma 3.1, one has 2c (k 2 (D n F ) k+2 ρ µ (M) = D n (F k ρ µ (M)) + ) cz µµ + d D(µ,µ) (F k ρ µ (M)) for all M = ( c d ) Γ 1. Here D (i,j) is the differential operator given by the determinant of the matrix D n with the i-th row and the j-th column removed. When n = 1, we let D (1,1) 0 = 1. (Proof ) For simplicity we can choose µ = 1. The differential operator D n can be written as D n = 2 τ D (1,1) ( 1) j vj D (1,j+1). where we have written the variable in H n as ( ) τ v t Z = (τ C, v C v z, z M, (C))

4 YOUNGJU CHOIE and we have used D (1,j+1) for the determinant of the matrix i,j with the 1-st row and the (j + 1)-th column removed. Using this decomposition we find for M Γ 1 D n (F k ρ 1 (M)) = D n ((cτ + d) k v F (Mτ, cτ + d, z c cτ + d vvt )) = 2 τ ((cτ + d) k D (1,1) F (ρ 1 (M)Z)) (cτ + d) k n j=2 ( 1) j 1,j D (1,j) (F (ρ 1 (M)Z))) = 2ck cτ + d D(1,1) (F k ρ 1 (M)) + (D n (F )) k+2 ρ 1 (M) c(cτ + d) k 2 v j ( vj D (1,1) F )(ρ 1 (M)Z) +c 2 (cτ + d) k 2 1 i,j v i v j ( zi,j D (1,1) F )(ρ 1 (M)Z) +(cτ + d) k ( 1) j vj D (1,j+1) (F (ρ 1 (M)Z))) 2c(k 2 = (D n F ) k+2 ρ 1 (M) ) D (1,1) (F k ρ 1 (M)), cτ + d because (cτ + d) k 1 ( 1) j D (1,j+1) (( vj F )(ρ 1 (M)Z)) c(cτ + d) k 1 1 i,j ( 1) j (1 + δ i,j )D (1,j+1) ((v i zi,j F )(ρ 1 (M)Z)) = c(cτ + d) k 2 v j (D (1,1) vj F )(ρ 1 (M)Z)) c(cτ + d) k 1 (n 1)(D (1,1) F )(ρ 1 (M)Z)) c 2 (cτ + d) k 2 1 i,j v i v j ( zi,j D (1,1) F )(ρ 1 (M)Z)). In general, the following lemma holds. Lemma 3.3. Let F be a holomorphic function on H n. Then, for each nonnegative integer l, each positive integer 1 µ n and ρ µ as in Lemma 3.1, one has (D l nf ) k+2l ρ µ (M) = with α n = k (Proof) 2. l From the fact that 1 D n ( cz µµ + d F (Z)) = (2c) ( ) l j l j (αn + l 1)! (D(µ,µ) (α n + j 1)!(cz µµ + d) l j ) l j D j n(f k ρ µ (M)), 2c (cz µµ + d) 2 D(µ,µ) (F (Z)) + for all M Γ n, and all µ n, we obtain the result by induction on l. 1 cz µµ + d D n(f (Z)) The following theorem is generalizing the idea of Eichler-Zagier given in [[7], I.3, pp.28 35]. This was done by using the exponential differential operators. This leads us how to

MULTILINEAR OPERATORS ON SIEGEL MODULAR FORMS OF GENUS 1 AND 2 5 construct Siegel modular forms of genus n using the determinant-differential operator D n. This kind of generalization has been already studied in the case of Jacobi forms using the heat operator in [2]. Theorem 3.4. Let Z H n and ρ µ be the same as before. Let F (Z; X) = l=0 χ l(z)x l be a formal power series in X satisfying, for each 1 µ n, and any T = T t M n n (Z), (3.1) (3.2) F (Z + T ; X) = F (Z) X F (ρ µ (M)Z; (cz µµ + d) ) 2 = (cz µµ + d) k e 2cX czµ,µ+dd (µ,µ) F (Z; X). Here, the exponential e XD(µ,µ) is regarded as a formal power series in X the coefficients of which are the differential operators on H (H 0 = {1}); namely, e XD(µ,µ) X = (D l (µ,µ) )l l 0. l! Also assume that χ l (Z) is holomorphic in H n (in case when n = 1, assume that χ ν (Z) is holomorphic at every cusp as well). Then, for each nonnegative integer ν, Ψ ν (Z) = ν ( 1) j D j n(χ ν j )(α n + 2ν j 2)! j!(α n + ν 2)!, α n = k n 1, 2 is in M k+2ν (Γ n ). Remark 3.5. (1) When n = 1, after normalizing X to the proportion of z 2, z C, we can obtain Theorem3.2 of Eichler, Zagier: The theory of Jacobi forms[7] as a special case. (2) The functional equations given in (3.1), (3.2) are equivalent to saying that χ l satisfies and χ l (Z + T ) = χ l (Z), T = T t M n n (Z) χ l (ρ µ (M)Z) = (cz µµ + d) k+2l for every µ N, µ n. l (2c) l j (D (µ,µ) ) l j χ j (Z) (l j)!(cz µµ + d) l j (Proof of Theorem3.4) Let S k be the set of formal power series of the form F (Z; X) = l 0 χ l(z)x l satisfying the functional equation given in (3.2). Define an operator D k := D n α n X X 2 X 2 Here α n = k 2. Then, one can show, by direct computation using Lemma3.2, that D k maps S k to S k+2 ; D k ((cz µµ + d) k e czµµ+dd 2c (µ,µ) F (ρ µ (M)Z; X (cz µµ + d) 2 ) = (cz µµ + d) k 2 e czµµ+dd 2c (µ,µ) ( D X k F )(ρ µ (M)Z; (cz µµ + d) ) 2

6 YOUNGJU CHOIE So, in terms of power series, we get D k ( l 0 χ l (Z)X l ) = l 0 {D n (χ l ) (l + 1)(α n + l)χ l+1 }X l. Iterating this formula ν times, we find, by induction on ν, that the composition map Dk e Dk+2 e... S k Sk+2 Sk+4... S Dk+2ν e k+2ν 2 Sk+2ν. maps l 0 χ l(z)x l to ν ( 1) j+ν Dn(χ j l+ν j ) ( ν j ) (l + ν j)!(l + α n + 2ν j 2)! X l l!(l + α n + 2ν 2)! l 0 and composing this with the map F (Z; X) F (Z; 0) gives ( 1)ν (α n+ν 2)!ν! (α n Ψ +2ν 2)! ν, with ν ( 1) j D j Ψ ν (Z) = n(χ ν j )(α n + 2ν j 2)!. j!(α n + ν 2)! This means that we get, for each 1 ν N and for all µ n, Ψ ν satisfying (Ψ ν k+2ν ρ µ (M))(Z) = Ψ ν (Z). It is also true that Ψ ν satisfies Ψ ν (Z + T ) = Ψ ν (Z) for all T = T t M n n (Z) because χ l (Z + T ) = χ l (Z). So, Lemma3.1 implies that Ψ ν is a Siegel modular form of genus n (the condition at the cusp, when n = 1, can be checked immediately from the expansion). We now show how to construct multilinear forms on the spaces of Siegel modular forms of genus 1 and 2. Here, we note that this construction only works for the case when genus n = 1 or 2 due to the special property of the determinant-differential operator D n. See Remark3.8 for more detail. Theorem 3.6. Assume that n = 1 or n = 2. For j = 1,.., q let F j be a Siegel modular form of weight k j and genus n. Define k = q i=1 k j, α n = k, and β 2 j = k j. Then for 2 each nonnegative integer ν, P q r j+p=ν ( q 2 n ) p (α n + 2ν p 2)! D p n p!(α n + ν 1)! ( q ) D r j n (F j )(Z) r j!(β j + r j 1)! is a Siegel modular form of weight k + 2ν and genus n. Remark 3.7. (1) The case when n = 1 and q = 1 has been discussed in [7]. (2) The case when n = 2 and q = 2 has been discussed in [3]. (Proof) Let, for each j = 1, 2,...q, Then F j (Z; X) satisfies and F j (ρ µ (M)Z; F j (Z; X) = l 0 D l n(f j ) l!(β j + l 1)! Xl. F j (Z + T ; X) = F j (Z; X), T = T t M n n (Z) X (cz µµ + d) 2 ) = (cz µµ + d) k j e 2cX czµµ+dd (µ,µ) Fj (Z; X), 1 µ n.

Let MULTILINEAR OPERATORS ON SIEGEL MODULAR FORMS OF GENUS 1 AND 2 7 H(Z; X) := ( F 1 F2.. F q )(Z; X) = q l=0 r 1 +r 2 +...+r q =l D r j n (F j ) r j!(β j + r j 1)! Xl. Then, when n = 1, or 2, we note that H(Z; X) satisfies two functional equations in (3.1) and (3.2); and (3.3) H(Z + T ; X) = H(Z; X), T = T t M n n (Z) X H(ρ µ (M)Z; (cz µµ + d) ) = (cz 2 µµ + d) k e 2cq 2 n X czµµ+d D (µ,µ) H(Z; X). because e XD(µ,µ) F e XD (µ,µ) G = e 2 XD 2 n (µ,µ) F G. So, by applying Theorem3.4 to H(Z; X), we obtain a Siegel modular form Ψ ν (Z), ν Ψ ν (Z) = ( 1) p (q 2 n ) p Dn( p p=0 q r 1 +..+r q =ν p D r j n (F j ) r j!(β j + r j 1)! )(α n + 2ν p 2)! p!(α n + ν 2)! of weight k + 2ν and genus n. Normalizing Ψ ν (Z) by the factor (α n + ν 1), we obtain the result. Remark 3.8. We note that the functional equation of H(Z, X) in (3.3) is valid only when n is 1 or 2. For holds only when n = 1 or 2. e XD(µ,µ) F e XD (µ,µ) G = e 2 XD 2 n (µ,µ) F G 4. Conclusion In this paper we consider multilinear differential operators on the space of Siegel modular forms of genus 1 and 2. To do that, we have introduced exponential differential operators. One can also construct multilinear differential operators on the space of Jacobi forms using generating function studied in [2]. It seems surprising that the method of using generating function, after introducing the exponential differential operators, is rather general. It might be very interesting to find the right differential operator which holds Leibnitz rule to get bilinear covariant differential operators on the space of Siegel forms with higher genus n > 2., References [1] Y. Choie, Jacobi forms and the heat operator, Math. Zeit., 225, (1997), No1, 95-101. [2] Y. Choie, Jacobi forms and the heat operatorii, Illinois Journal of Mathematics, Vol.42, Summer (1998), No.2, 179-186. [3] Y. Choie and W. Eholzer, Rankin-Cohen operator and Jacobi and Siegel forms, Jour. of Number Theory, Vol.68, (1998), No.2, 160-177. [4] H. Cohen, Sums involving the values at negative integers of L functions of quadratic characters, Math. Ann., 217, (1975), 271-285. [5] P. Cohen, Y. Manin and D. Zagier, Automorphic Pseudodifferential operators, Algebraic aspects of integrable systems, Fokas and Gelfand, editors; Progress in Nonlinear Differential Equations and their Applications, 26: Birkhäuser(1997). [6] W. Eholzer: Private communication (1996).

8 YOUNGJU CHOIE [7] M. Eichler and D. Zagier, The Theory of Jacobi Forms, (Prog. Math, Vol. 55 ) Boston Basel Stuttgart: Birkhäuser (1985). [8] H. Maass, Siegel s modular forms and Dirichlet series, Lecture Notes in Math. 216, (1971), Springer- Verlag. [9] R. Rankin, The construction of automorphic forms from the derivatives of a given form, J. Indian Math. Soc, (1956), 20, 103-116. [10] R. Rankin, The construction of automorphic forms from the derivatives of given forms, Michigan Math. J. 4,(1957), 181-186. [11] A. Unterberger and J. Unterberger, Algebras of symbols and modular forms, Jour. d Analyse Math., 68, (1996), 121-143. [12] D. Zagier: Modular forms and differential operators, Proceedings of the Indian Academy of Sciences, Vol.104, (1994), No.1, 57-75. [13] D. Zagier, Introduction to modular forms, From Number Theory to Physics, M.Waldschmidt, P. Moussa, J.M. Luck, C. Itzykson, editors; Springer-Verlag, (1992), 238-291. Department of Mathematics, Pohang University of Science and Technology, Pohang, 790 784, Korea E-mail address: yjc@vision.postech.ac.kr, yjc@yjc.postech.ac.kr