Abh. Math. Sem. Univ. Hamburg 65 (1995), 89-93

A Note on Jacobi Forms of Higher Degree

BY K. CHAKRABORTY, B. RAMAKRISHNAN, and T.C. VASUDEVAN

1 Introduction

In his paper [1], M. EICHLER proved, using the Fourier Jacobi expansion of a Siegel modular form, that the space of Siegel modular form of degree n < 4 for some specific weights k is one-dimensional. In the present note, using a similar method, we prove that the dimension of the space of Jacobi forms of degree 2 < n < 5 with index # for some specific weights is less than or equal to the dimension of the space of Jacobi forms of degree 1, index # with the corresponding weights.

2 Preliminaries

We follow mostly the notations as in [4]. Let us denote by ~, R and Z, the field of complex numbers, the field of real numbers and the ring of rational integers respectively. Let ~n denote the Siegel upper half-space of degree n E ]N. Let F, = Sp(n,Z) be the symplectic group of degree n. Let k E N and let G~ '1) = Sp(n,7Z) • H~ "1) be the full Jacobi group of degree n, where H (n'l)

is the Heisenberg group (here we consider the case j = 1 in ZIEGLER'S paper [4]). Let/~ be a natural number. Denote by Jk,~(F,), the space of holomorphic Jacobi forms of degree n, weight k with index # for the group G~ '1).

3 The results

Let Z = (Z1 C ~n-1 0 )

0 zn C Jg~l C ~ n .

Put W = (W1, wl), W1 E IE ~1'"-1), wl E IE. Then the Fourier-Jacobi expansion of ~(Z, W) ~ Jk,~(Fn) can be written as

(I)(Z, W) = Z Wm,rl (Zl, W1) e2nimzne2nirlwl (1) m>_O,rl c7l

(see [4], p. 215.) Note that tPm,rl (Z1, W1) C Jk,~(Fn-l).

90 K. Chakraborty, B. Ramakrishnan, and T.C. Vasudevan

Remark I. When n = 1, Jk,~(F1) is the usual Jacobi form of degree I as in [2].

Definition. We define the order of ~(Z, W) to be the least integer m in the expansion (1) with the property that ~Pm,r~ =P 0.

We first prove

Theorem 1.

where

2 m < ~ - ~ k

Using Theorem 1, we prove the main theorem of this note in

(2)

Theorem 2. The dimension of the space Jk.u(Fn) over ~, in the following cases, is less than or equal to the dimension over �9 of the space o f Jacobi forms of degree 1, weight k with index #.

i) n = 2 , k = 4 , 6, 8.

ii) n = 3, k - - 4 , 6.

iii) n = 4 , k = 4 .

iv) n = 5 , k = 4 .

Remark 2. If the Siegel-Jacobi operator 9 ~ (which is a map from Jg,~(F,) to Jk,~(Fn-1) defined on page 199 of [4]) is onto in the respective cases of Theorem 2, then we will get the equality in Theorem 2. Also if Jk,u(Ft) for k = 4, 6, 8 is of one dimensional, then the space Jk,~,(F,) (for the above cases of Theorem 2) is also of one dimensional. (Of course, it is so if # = 1.)

4 Proofs

Since we mainly follow the proof of EICHLER, we only sketch the proofs. Put

g(Z, W) = (det y)ke-4~u(~Y-~lJ')I~(Z, W)[ 2

where Z = X + iY ; W = ~ + ifl. Then (g(Z,W)) 1/2 is G~'l)-invariant and is bounded. Therefore, there exists (Z', W') lying in a suitably chosen fundamental domain of G~ '1), where g attains its maximum (say) M. Write Z I = X I + iY' and W' = a' + ifl'.

Now choose a vector c ~ 0 such that y,-1 [c] is minimal (here M[X] stands for XtMX) . Also there exists a constant #, depending only on n such that

2 2 0 < Y'-l[c] <_ #.(det yt-1)l/n < ~ # n (3)

A Note on Jacobi Forms of Higher Degree 91

where #, < (2/x/~) "-t is the well-known Hermite's estimate. Put C = yI c t y I - l c . Define

rTr l \ --2nimzn ~o(z,) = O(Z' + z,c, w )e

Then we obtain

i~(0)l 2 ~ max I~(z.) l 2 (1 - e 2 n y . ) 2 '

zn = x, + iyn. (4)

Now multiplication by e-4~(fl'Y'-lfl")(det y,)k on the left hand side of (4) will give g(Z', W') which is equal to M. On the otherhand, multiplication by det(Y' + y,C)ke -4~"y" on the right hand side will give atmost the maximum of g(Z, W) which is M. Therefore we will get the following estimate

e -4nmyn det(Y') k det(Y' + ynC) -k 1 - - e 2~zy" ) 2

i . e,~

4rim < y,-I [c]k + o(y~)k + __2_2 [ log(1 - e2ny")l. (5) - l Y . I

Now, if one considers the /-th power of @(Z, W), both the order m and the weight k will be multiplied by l. Therefore, considering the equation (5) in that case and allowing l ~ oo and y, ~ 0, we will get

4nm < y,-1 [c]k.

Theorem 1 now follows from the above equation by using (3).

Remark 3. Actually, there exists an improved estimate of (3) given by

0 < Y'-l[c] _< hn(det y,-1)1/~ < __~3hn2

where the optimal values of the constants hn for n < 10 are well known. (for this fact one refers [3], p. 50). In fact, h5 = 81/5. i.e., we have

4~zm < --~3 hn2 k

o r

hn2 k m _< 2 v ~ ' (6)

Proof of Theorem 2. Let r be the dimension of Jk,u(F1) over I13. We will prove the result for n = 2 and the proof follows by induction. If possible choose r + 1 linearly independent elements (over ~) in Jk,~(U2) (say) @1,'",@r+1,

92 K. Chakraborty, B. Ramakrishnan, and T.C. Vasudevan

where k = 4, 6, 8. The idea is to prove that there exists a linear combination of the ~i 's which has order m > 1 to get a contradiction to (2).

Using (1), the Fourier-Jacobi expansion of each q)i can be written as

(~i(Z, W) = E lPm,rli (Z1, W1)e2nimZ"e2nirlwl. m>_O,rt E77

Note that rl = 0 if m -= 0. Consider the r + 1 elements ~P~,0 E Jk,u(F0, i = 1 , . . . , r + 1. Since Jk,u(F~)

is r-dimensional over r there exists constants ~i ~ ll~, i = 1 , . . . , r + 1 not all zero such that

r+l

= 0 i=1

which implies that r+l

~'(Z, W) = ~ ~i~i(Z, W) i=l

has order m > 1. Therefore using (2), we get k > 8, a contradiction. Note that for the case n = 5, we have to use (6) to get a contradiction. This completes the proof. []

Acknowledgements

The authors wish to thank Professor ALOYS KRIEG and the referee for making some crucial remarks. The second author acknowledges the N B H M grant. He also acknowledges the warm hospitality of the Fields Institute, Waterloo, Canada, where he had discussions with Professor KRIEG during his recent visit.

References

[1] M. EICHLER. Ober die Anzahl der linear unabh~ingigen Siegelschen Modul- formen von gegebenem Gewicht. Math. Ann. 213 (1975), 281-291. "Erratum". ibid. 215 (1975), 195.

[2] M. EICnLER and D. ZAGIER. The Theory of Jacobiforms. Progress in Math. 55. Birkhiiuser, Boston 1985.

[3] E. FREITAG. Siegelsche Modulfunktionen. Grundl. Math. Wiss. 254 1983.

[4] C. ZIEGLER. Jacobi forms of higher degree. Abh. Math. Sere. Univ. Hamburg 59 (1989), 191-224.

Eingegangen am." 09.07.1993 in revidierter Fassung am: 31.05.1994

A Note on Jacobi Forms of Higher Degree 93

Authors' addresses: K. Chakraborty, B. Ramakrishnan, Mehta Research Institute of Mathematics and Mathematical Physics, 10, Kasturba Gandhi Marg (Old Kutchery Road), Allahabad 211 002, India.

T.C. Vasudevan, RKM Vivekananda College, Mytapore, Madras 600 004, India.