Math. J. Okayama Univ. 63 (2021), 123–131

DIFFERENTIAL OPERATORS ON MODULAR FORMSASSOCIATED TO JACOBI FORMS
Min Ho LeeAbstract. Given Jacobi forms, we determine associated Jacobi-likeforms, whose coefficients are quasimodular forms. We then use thesequasimodular forms to construct differential operators on modular forms,
which are expressed in terms of the Fourier coefficients of the given Jacobi forms.

1. Introduction
Jacobi forms were introduced by Eichler and Zagier in [4], and they play
an important role in number theory. Jacobi-like forms are formal power
series with coefficients in the ring of holomorphic functions on the Poincaré
upper half plane H which are invariant under a certain action of a discrete
subgroup of SL(2, R), and they generalize Jacobi forms in some sense (cf.
[3], [8]). On the other hand, quasimodular forms, which were introduced
by Kaneko and Zagier in [5], generalize modular forms, and they are closely
linked to Jacobi-like forms. More specifically, each coefficient of a Jacobi-likeform is a quasimodular form, and there is a lifting map from quasimodularforms to Jacobi-like forms so that the lifting of a quasimodular form has thisform as one of its coefficients (cf. [1], [6]).
Although the derivative of a modular form is not a modular form ingeneral, there are a number of ways of constructing differential operators
on modular forms. For example, the Serre operator can be considered byusing the Eisenstein series E2 , and Rankin-Cohen brackets determine certain
types of differential operators on modular forms. The goal of this paper isto construct differential operators on modular forms associated a Jacobiform by using a method of constructing such operators from quasimodularforms introduced in our earlier paper [7]. To be more specific, we firstdetermine Jacobi-like forms corresponding to Jacobi forms, so that theircoefficients are quasimodular forms. We then use these quasimodular forms
to construct differential operators on modular forms by using the abovementioned method. These operators are expressed in terms of the Fourier
coefficients of the given Jacobi forms.
Mathematics Subject Classification. Primary 11F11; Secondary 11F50.
Key words and phrases. Jacobi forms, Jacobi-like forms, modular forms, quasimodularforms.
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M. H. LEE

2. Jacobi forms and Jacobi-like forms
Let H be the Poincaré upper half plane on which the group SL(2, R) acts
as usual by linear fractional transformations. Thus we may write
γz =for all z ∈ H and γ =(2.1)

a bc d

az + bcz + d∈ SL(2, R). For the same z and γ, we setJ(γ, z) = cz + d,K(γ, z) =c.
cz + dThen the resulting maps J, K : SL(2, R) × H → C satisfy
J(γγ ′ , z) = J(γ, γ ′ z)J(γ ′ , z),K(γ, γ ′ z) = J(γ ′ , z)2 (K(γγ ′ , z) − K(γ ′ , z))

for all γ, γ ′ ∈ SL(2, R) and z ∈ H.
We now consider a subgroup Γ of SL(2, Z) of finite index and choose
nonnegative integers ν and µ. Then Jacobi forms for Γ introduced by Eichler
and Zagier [4] can be described as follows.
Definition 2.1. A holomorphic function φ : H × C → C is a Jacobi formof weight ν and index µ for Γ if it satisfies the following conditions:(i) If J and K are as in (2.1), the transformation formula(2.2)

2φ(γz, J(γ, z)−1 w) = J(γ, z)ν e2πiµK(γ,z)w φ(z, w)

holds for all γ ∈ Γ and (z, w) ∈ H × C.
(ii) If (p, q) ∈ Z2 , the relation
(2.3)

2

φ(z, w + pz + q) = e2πiµ(−p z−2pw) φ(z, w)

holds for all (z, w) ∈ H × C.
(iii) The function φ has a Fourier development of the form
(2.4)

φ(z, w) =∞XXCφ (n, r)e2πinz/ℓ e2πirw

n=0 r 2 ≤4µn/ℓ

for all (z, w) ∈ H × C, where ℓ is the least positive integer such that φ(z +ℓ, w) = φ(z, w).
We denote by Jν,µ (Γ) the space of Jacobi forms of weight ν and index µfor Γ.
Remark 2.2. Given a Jacobi form φ ∈ Jν,µ (Γ), we set φ− (z, w) = φ(z, −w).
Then for γ ∈ Γ, (z, w) ∈ H × C and (p, q) ∈ Z2 , using (2.2) and(2.3), weDIFFERENTIAL OPERATORS ON MODULAR FORMS

125haveφ− (γz, J(γ, z)−1 w) = φ(γz, J(γ, z)−1 (−w))
2= J(γ, z)ν e2πiµK(γ,z)w φ(z, −w)
2= J(γ, z)ν e2πiµK(γ,z)w φ− (z, w),
φ− (z, w + pz + q) = φ(z, (−w) + (−p)z + (−q))
2= e2πiµ(−p z−2pw)φ(z, −w)2= e2πiµ(−p z−2pw)φ− (z, w).
Thus it follows that φ− is also a Jacobi form belonging to Jν,µ (Γ).
Let F be the ring of holomorphic functions f : H → C that are polynomially bounded, meaning that there is a constant N > 0 such thatf (z) = O((1 + |z|2 )/y N )for z ∈ H with y = Im(z) as y → ∞ and y → 0 (see e.g. [2, §5.1]). We denoteby F[[X]] the complex algebra of formal power series in X with coefficients
in F, so that its elements can be written in the form
Φ(z, X) =∞Xφk (z)X kk=0with φk ∈ F for each k ≥ 0. Given an integer λ, we can consider rightactions |λ and |Jλ of SL(2, R) on F and F[[X]], respectively, defined by(f |λ γ)(z) = J(γ, z)−λ f (z)(2.5)(Φ |Jλ γ)(z, X) = J(γ, z)−λ e−K(γ,z)X Φ(γz, J(γ, z)−2 X),for f ∈ F, Φ(z, X) ∈ F[[X]], γ ∈ SL(2, R) and z ∈ H.
Definition 2.3. (i) A holomorphic function f ∈ F is a modular form ofweight λ for Γ if it satisfies
f |λ γ = f
for all γ ∈ Γ.
(ii) A Jacobi-like form of weight λ for Γ is a formal power series Φ(z, X) ∈F[[X]] satisfying(2.6)
for all z ∈ H and γ ∈ Γ.

(Φ |Jλ γ)(z, X) = Φ(z, X)126

M. H. LEE

We denote by Mλ (Γ) and Jλ (Γ) the spaces of modular forms and Jacobilike forms, respectively, of weight λ for Γ.
We now fix a Jacobi form φ ∈ Jν,µ (Γ), which has a Fourier development asin (2.4), and consider the associated formal power series Fφ (z, X) ∈ F[[X]]defined by∞
XFφ (z, X) =ξkφ (z)X k ,(2.7)

k=0where(2.8)∞(2πi)k X X
φr 2k Cφ (n, r)e2πinz/ℓξk (z) =k(2k)!µ n=0 2r ≤4µn/ℓfor k ≥ 0.
Proposition 2.4. The formal power series Fφ (z, X) in X given by (2.7) isa Jacobi-like form belonging to Jν (Γ).
Proof. From (2.4) we see that the power series expansion of the Jacobi formφ(z, w) in w can be written as
φ(z, w) =∞Xβkφ (z)wk ,k=0

where(2.9)

βkφ (z) =∞XX

n=0 r 2 ≤4µn/ℓ(2πir)kCφ (n, r)e2πinz/ℓk!

for k ≥ 0. We consider the corresponding even series given by∞(2.10)

X φb w) = φ(z, w) + φ(z, −w) =φ(z,β2k (z)w2k .
2k=0

On the other hand, if we set
X = 2πiµw2 ,DIFFERENTIAL OPERATORS ON MODULAR FORMS

127

from (2.7), (2.8), (2.9) and (2.10) we obtain(2.11)Fφ (z, X) =∞∞X(2πi)k X
k=0=∞X
k=0(2k)!µk

Xr 2k Cφ (n, r)e2πinz/ℓ X k

n=0 r 2 ≤4µn/ℓ∞

X X
1(2πiµ)k2

n=0 r ≤4µn/ℓ(2πir)2k
Cφ (n, r)e2πinz/ℓ X k(2k)!


∞
∞
X
X k X φφ=β2k (z)w2k=β2k (z)2πiµk=0k=0b w).
= φ(z,Since φ is a Jacobi form of weight ν and index µ, by Remark 2.2 the sameb and therefore by (2.2) it satisfies
is true for φ,2bb w)
φ(γz,J(γ, z)−1 w) = J(γ, z)ν e2πiµK(γ,z)w φ(z,for all γ ∈ Γ and (z, w) ∈ H × C. From this and (2.11)we see thats
X−2−1Fφ (γz, J(γ, z) X) = φb γz, J(γ, z)
2πiµ
 sX
= J(γ, z)ν eK(γ,z)X φb z,2πiµ= J(γ, z)ν eK(γ,z)X Fφ (z, X);

hence Fφ (z, X) belongs to Jν (Γ).



3. Quasimodular forms and differential operators
Let Γ and F be as in Section 2, and let m and λ be integers with λ ≥2m ≥ 0.
Definition 3.1. An element f ∈ F is a quasimodular form for Γ of weight
λ and depth at most m if there are functions f0 , . . . , fm ∈ F satisfying(3.1)(f |λ γ)(z) =m
X

fr (z)K(γ, z)r

r=0for all z ∈ H and γ ∈ Γ, where K(γ, z) is as in (2.1). We denote by QMλm (Γ)
the space of such quasimodular forms.
Remark 3.2. As was pointed out by Zagier in [9, Section 5.3] and [10,Section 2], the condition for quasimodular forms given in Definition 3.1 wassuggested by Werner Nahm.

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M. H. LEE

We note that the functions fr in (3.1) are uniquely determined by f andthat they are in fact quasimodular forms such thatm−r
fr ∈ QMλ−2r(Γ)

for 0 ≤ r ≤ m (see e.g. [1]). Thus by setting(3.2)

Sr (f ) = fr ,we obtain the complex linear mapm−r
Sr : QMλm (Γ) → QMλ−2r(Γ)

for each r.
Quasimodular forms are closely linked to Jacobi-like forms. Indeed, if theformal power series∞X(3.3)Ψ(z, X) =ψk (z)X k .
k=0

is a Jacobi-like form belonging to Jν (Γ), then the coefficient function ψmmwith m ≥ 0 is a quasimodular form belonging to QMν+2m(Γ). Thus we canconsider the complex linear map
(3.4)

m(Γ)
Πm : Jν (Γ) → QMν+2m

defined byΠm (Ψ(z, X)) = ψmmfor Ψ(z, X) ∈ Jν (Γ) as in (3.3). Furthermore, if Sj : QMν+2m(Γ) →m−j
QMν+2m−2j (Γ) is as in (3.2), using Corollary 3.7 and Proposition 3.10 in[1], we have1(3.5)
Sj (ψm ) = ψm−jj!
for 0 ≤ j ≤ m. The next proposition provides a method of determining a
differential operator on modular forms associated to a quasimodular form.
Proposition 3.3. Given a quasimodular form ψ ∈ QMλm (Γ) and nonnegative integers u and σ with u ≤ m, there is a differential operatorAσψ,u : Mσ (Γ) → Mσ+λ−2m+2u (Γ)
on modular forms given byu XsX(−1)s (m − u + s)!(2u + σ + λ − 2m − s − 2)!
σAψ,u =(3.6)
p!(s − p)!
s=0 p=0× (Sm−u+s ψ)(s−p)
Proof. See [7].

dp.
dz p

DIFFERENTIAL OPERATORS ON MODULAR FORMS129

We now consider a Jacobi form φ ∈ Jν,µ (Γ) whose Fourier coefficientsCφ (n, r) ∈ C are as in (2.4), and set(3.7)

φ,u,σKn,p=uXX

s=p r 2 ≤4µn/ℓ(−1)s (2πi)u−p r 2u−2s ns−pµu−s ℓs−p×(2u + σ + ν − s − 2)!
Cφ (n, r).
p!(s − p)!(2u − 2s)!

for nonnegative integers k, p, σ and u with p ≤ u. We then define theassociated functions
Fpφ,u,σ : H → Cby the series
Fpφ,u,σ (z) =∞Xφ,u,σ 2πinz/ℓeKn,pn=0

for all z ∈ H. The differential operator on modular forms associated to φ isthen described in the next theorem.
Theorem 3.4. Given a Jacobi form φ ∈ Jν,µ (Γ) and nonnegative integers
u and σ, the formulaDφ,u,σ =uX

Fpφ,u,σ (z)p=0dpdz pdetermines a differential operator
Dφ,u,σ : Mσ (Γ) → Mσ+ν+2u (Γ)
on modular forms for Γ.
Proof. Let Fφ (z, X) be the formal power series given by (2.7), which belongs
to Jλ (Γ) by Proposition 2.4. If Πm with m ≥ 0 is the map in (3.4), then we
see that the functionφξm= Πm (Fφ (z, X))m
is a quasimodular form belonging to QMν+2m(Γ), which satisfies
φSj (ξm)=1 φξ
j! m−jfor 0 ≤ j ≤ m by (3.5). From this relation and (2.8) we obtain(3.8)

φSj (ξm)=∞XX

n=0 r 2 ≤4µn/ℓ(2πi)m−j r 2m−2j
Cφ (n, r)e2πinz/ℓ .
j!(2m − 2j)!µm−j

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M. H. LEE

We now apply Proposition 3.3 by using λ = ν + 2m in (3.6) and setting
Dφ,u,σ = Aσφ to obtain the operator
ξm ,uDφ,u,σ : Mσ (Γ) → Mσ+ν+2u (Γ)
given byDφ,u,σ =u X
sX(−1)s (m − u + s)!(2u + σ + ν − s − 2)!

p!(s − p)!

s=0 p=0p
φ (s−p) d.
× (Sm−u+s ξm )pdz

On the other hand, using (3.8), we haveφ)(z) =(Sm−u+s ξm∞XX

n=0 r 2 ≤4µn/ℓ(2πi)u−s r 2u−2s
Cφ (n, r)e2πinz/ℓu−s(m − u + s)!(2u − 2s)!µfor 0 ≤ s ≤ u ≤ m, so that∞XX(2πi)u−s r 2u−2s(m − u + s)!(2u − 2s)!µu−s
n=0 r 2 ≤4µn/ℓ
 2πin s−p
Cφ (n, r)e2πinz/ℓ×ℓfor 0 ≤ p ≤ s. Thus we obtainφ (s−p)(Sm−u+s ξm)(z) =Dφ,u,σ =u Xs X∞XX

s=0 p=0 n=0 r 2 ≤4µn/ℓ(−1)s (2u + σ + ν − s − 2)!
p!(s − p)!

p(2πi)u−p r 2u−2s  n s−p2πinz/ℓ dC(n,r)eφ(2u − 2s)!µu−s ℓdz pu Xs X∞
X
X (−1)s (2πi)u−p (2u + σ + ν − s − 2)!r 2u−2s ns−p×=s=0 p=0 n=0 r 2 ≤4µn/ℓ

p!(s − p)!(2u − 2s)!µu−s ℓs−pp
2πinz/ℓ d× Cφ (n, r)edz p=u X∞u XXX

p=0 s=p n=0 r 2 ≤4µn/ℓ(−1)s (2πi)u−p (2u + σ + ν − s − 2)!r 2u−2s ns−pp!(s − p)!(2u − 2s)!µu−s ℓs−pdpdz puu∞pXXXdp
φ,u,σφ,u,σ 2πinz/ℓ d=,F(z)=Kn,pepppdz
dz
p=0
p=0 n=0× Cφ (n, r)e2πinz/ℓDIFFERENTIAL OPERATORS ON MODULAR FORMS

where we used (3.7); hence the proof of the theorem is complete.

131

References[1] Y. Choie and M. H. Lee, Quasimodular forms and Jacobi-like forms, Math. Z. 280(2015), 643–667.
[2] H. Cohen and P. F. Strömberg, Modular forms, Amer. Math. Soc., Providence, 2017.
[3] P. B. Cohen, Y. Manin, and D. Zagier, Automorphic pseudodifferential operators,Algebraic Aspects of Nonlinear Systems, Birkhäuser, Boston, 1997, pp. 17–47.
[4] M. Eichler and D. Zagier, The theory of Jacobi forms, Progress in Math., vol. 55,
Birkhäuser, Boston, 1985.
[5] M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodularforms, Progress in Math., vol. 129, Birkhäuser, Boston, 1995, pp. 165–172.
[6] M. H. Lee, Cohen-Kuznetsov liftings of quasimodular forms, Acta Arith. 171 (2015),241–256.
[7] M. H. Lee, Differential operators on modular forms associated to quasimodular forms,Ramanujan J. 39 (2016), 133–147.
[8] D. Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. Math.
Sci. 104 (1994), 57–75.
[9] D. Zagier, Elliptic modular forms and their applications, In: Ranestad, K., Bruinier,J.H., Harder, G., van der Geer, G., Zagier, D. (eds.) The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway. Universitext, pp. 1–103. Springer,Berlin (2008)[10] D. Zagier, Partitions, quasimodular forms, and the Bloch-Okounkov theorem, Ramanujan J. 41 (2016), 345–368.
Min Ho Lee
Department of Mathematics
University of Northern Iowa
Cedar Falls, Iowa 50613, U. S. A.
e-mail address: lee@math.uni.edu(Received April 4, 2019 )(Accepted November 8, 2019 )