jacobi-like forms, differential equations, and hecke operators
TRANSCRIPT
This article was downloaded by: [University of Tennessee At Martin]On: 05 October 2014, At: 07:52Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Complex Variables, Theory andApplication: An International JournalPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gcov19
Jacobi-like forms, differentialequations, and Hecke operatorsMin Ho Lee aa Department of Mathematics, University of Northern Iowa, CedarFalls, IA 50614, USAPublished online: 17 Feb 2007.
To cite this article: Min Ho Lee (2005) Jacobi-like forms, differential equations, and Heckeoperators, Complex Variables, Theory and Application: An International Journal, 50:14, 1095-1104
To link to this article: http://dx.doi.org/10.1080/02781070500324300
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Complex VariablesVol. 50, No. 14, 15 November 2005, 1095–1104
Jacobi-like forms, differential equations,and Hecke operators
MIN HO LEE*
Department of Mathematics, University of Northern Iowa,Cedar Falls, IA 50614, USA
Communicated by R. P. Gilbert
(Received in final form 17 April 2005)
We construct a map from the space of Jacobi-like forms J (�) for a discrete subgroup�� SLð2,RÞ to the space
Q1m¼1 Pm, ’ of sequences of meromorphic functions satisfying certain
conditions determined by some linear ordinary differential operators and prove that theHecke operator actions on J (�) and on
Q1m¼1 Pm, ’ are compatible with respect to this map.
Keywords: Jacobi-like forms; Modular forms; Hecke operators; Differential equations
2000 Mathematics Subject Classifications: 11F11; 11F50; 34M15
1. Introduction
Jacobi forms generalize theta functions, and they occur naturally in number theoryas Fourier coefficients of Siegel modular forms, for example. A systematic study ofJacobi forms was introduced by Eichler and Zagier in [2]. Jacobi-like forms arepower series satisfying a certain transformation formula under the operation of adiscrete subgroup � of SLð2,RÞ, which is essentially one of the two transformationformulas satisfied by Jacobi forms for �. It is known that there is a correspondencebetween Jacobi-like forms and certain sequences of modular forms [1]. Indeed, thetransformation formula for a Jacobi-like form determines an expression of each ofits coefficients in terms of derivatives of some modular forms belonging to theassociated sequence. In light of their intimate relations with Jacobi and modularforms, it is not surprising that Jacobi-like forms are closely linked to number theory.Recently, however, they have also been studied in connection with other areassuch as the theory of vertex algebras and conformal field theory [6].
*Email: [email protected]
Complex Variables
ISSN 0278-1077 print: ISSN 1563-5066 online � 2005 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/02781070500324300
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
Another interesting source of sequences of modular forms of various weights comesfrom the theory of linear ordinary differential equations [7]. Given a meromorphicmodular form ’(z) of weight one and a positive integer m, there is a linearordinary differential operator SmD’ of order ðmþ 1Þ such that the functionsfzm�i’ðzÞm j 1 � i � mg is a set of linearly independent solutions of the homogeneousequation SmD’ f ¼ 0. If ’,m is a meromorphic function on H satisfying a certaincondition determined by SmD’ (see section 4) and if Sð ’,mÞ is a solution of the differ-ential equation SmD’ f ¼ ’,m, then the ð2mþ 1Þ-st derivative of Sð ’,mÞ’
�2m is amodular form of weight 2mþ 2. Sequences of such modular forms can be used toconstruct Jacobi-like forms.
One of the effective tools in the theory of modular forms is the notion of Heckeoperators. Hecke operators can also be considered in various other cases, where anaction of a discrete subgroup of a semisimple Lie group is involved. In particular,Hecke operators on Jacobi-like forms were introduced in [4]. If
Q1m¼1 Pm, ’ denotes
the space of sequences ð m, ’Þ1m¼1 of meromorphic functions m,’ described above,
Hecke operators acting on this space were studied in [3].In this article, we construct a map from the space of Jacobi-like forms J (�) for �
to the spaceQ1
m¼1 Pm, ’ and prove that the Hecke operator actions on J (�) andon
Q1m¼1 Pm, ’ are compatible with respect to this map.
2. Hecke operators on modular forms
Let GLþð2,RÞ be the group of 2� 2 real matrices of positive determinant, and let
H ¼ z 2 C j Im z > 0f g
be the Poincare upper half plane. Then GLþð2,RÞ acts on H by linear fractionaltransformations. Thus, if � ¼ ð ac
bd Þ 2 GLþð2,RÞ, we have
�z ¼azþ b
czþ d
for all z 2 H. Given such � and an integer k, we set
jð�, zÞ ¼ czþ d ð2:1Þ
for z 2 H. Then this determines the map j : GLþð2,RÞ � H satisfying the cocyclecondition
jð��0, zÞ ¼ jð�,�0zÞjð�0, zÞ ð2:2Þ
for all �,�0 2 GLþð2,RÞ and z 2 H. If f : H! C is a function, for each � 2 GLþð2,RÞand an integer k we denote by f jk � the function on H defined by
f jk �ð ÞðzÞ ¼ detð�Þk=2jð�, zÞ�kfð�zÞ
1096 M. H. Lee
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
for all z 2 H. Using (2.2), it can be shown that
f jk ð��0Þ ¼ ð f jk �Þ jk �
0 ð2:3Þ
for all �, �0 2 GLþð2,RÞ.Let �� SLð2,RÞ be a Fuchsian group of the first kind, where SLð2,RÞ is the
subgroup of GLþð2,RÞ consisting of the matrices of determinant one. This meansthat � is a discrete subgroup of SLð2,RÞ such that the quotient space �nH� is compact,where H� denotes the union of H and the set of cusps of �.
Definition 2.1 Given k 2 Z, a meromorphic modular form of weight k for �is a meromorphic function f : H! C which is meromorphic at the cusps of � andsatisfies
f jk � ¼ f
for all � ¼ ð acbd Þ 2 �: We denote by Mkð�Þ the space of all meromorphic modular
forms of weight k for �.
Hecke operators acting on Mkð�Þ can be described as follows. Two subgroups �1
and �2 of GLþð2,RÞ are said to be commensurable if �1 \ �2 has finite indexin both �1 and �2, in which case we write �1 � �2. Given a subgroup � ofGLþð2,RÞ, its commensurator ~�� � GLþð2,RÞ is given by
~�� ¼ g 2 GLþð2,RÞ j g�g�1 � �� �
:
If � � SLð2,RÞ is the Fuchsian group of the first kind considered above, the doublecoset ��� with � 2 ~�� has a decomposition of the form
��� ¼asi¼1
��i ð2:4Þ
for some �i 2 GLþð2,RÞ with i ¼ 1, . . . , s (see e.g. [5]). Given such � 2 ~�� andan element f 2 Mkð�Þ, we define the function Tkð�Þ f on H by
Tkð�Þ f ¼Xsi¼1
ð f jk �iÞ: ð2:5Þ
Using (2.3) and the decomposition (2.4), it can be shown that Tkð�Þf is alsoan element ofMkð�Þ.
Definition 2.2 The Hecke operator onMkð�Þ for k 2 Z associated to the double coset��� with � 2 ~�� is the linear map Tkð�Þ :Mkð�Þ !Mkð�Þ defined by (2.5).
Jacobi-like forms, differential equations, and Hecke operators 1097
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
3. Hecke operators on Jacobi-like forms
Let � � SLð2,RÞ be a Fuchsian group of the first kind as in section 2. In this sectionwe review Jacobi-like forms for � as well as the Hecke operators on the spaceof such Jacobi-like forms introduced in [4].
Let R be the ring of meromorphic functions on the Poincare upper half plane H,and let R½½X �� be the complex vector space consisting of all formal power seriesof the form �ðz,X Þ ¼
P1k¼0 �kðzÞX
k with �k 2 R for each k� 0.
Definition 3.1 A formal power series �ðz,X Þ 2 R½½X �� is a Jacobi-like form for � if
�ð�z, ðczþ d Þ�2X Þ ¼ ecX=ðczþd Þ�ðz,X Þ ð3:1Þ
for all z 2 H and � ¼ ð acbd Þ 2 �: We shall denote by J (�) the space of all Jacobi-like
forms for �.
Let �ðz,X Þ be a Jacobi-like form in J (�) of the form
�ðz,X Þ ¼X1k¼u
�kðzÞXk, ð3:2Þ
where u is a nonnegative integer. Then by (3.1) we have
X1k¼u
�kð�zÞðczþ d Þ�2kXk ¼X1k¼0
1
k!
c
czþ d
� �k
Xk
! X1k¼u
�kðzÞXk
!:
Comparing the coefficients of X k, we see that a formal power series �ðz,X Þ 2 R½½X ��given by (3.2) is a Jacobi-like form for � if and only if its coefficients satisfy
ð�k j2k �ÞðzÞ ¼Xk�u‘¼0
1
‘!
c
czþ d
� �‘�k�‘ðzÞ
for all k � u and � ¼ ð acbd Þ 2 �:
PROPOSITION 3.2 A formal power series �ðz,X Þ 2 R½½X �� given by
�ðz,X Þ ¼X1k¼u
�kðzÞXk
is an element of J (�) if and only if there is a sequence f f‘g1‘¼u of modular forms
with f‘2M2‘ð�Þ for each ‘ such that
�k ¼Xk�ur¼0
1
r!ð2k� r� 1Þ!fðrÞk�r ð3:3Þ
for all k � u, where fðrÞk�r denotes the derivative of fk�r of order r.
Proof This can be shown by slightly modifying the proof of Proposition 2 in [1]. g
1098 M. H. Lee
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
Given an element � ¼ ð acbd Þ 2 GLþð2,RÞ and a formal power series �ðz,X Þ 2 R½½X ��,
we set
ð� j �Þðz,X Þ ¼ e�cX=jð�, zÞ�ð�z, ðdet �Þjð�, zÞ�2X Þ ð3:4Þ
for all z 2 H, where jð�, zÞ is as in (2.1). Thus, if � is a subgroup ofSLð2,RÞ � GLþð2,RÞ, the formal power series �ðz,X Þ is a Jacobi-like form for �if and only if � j � ¼ � for all � 2�. Furthermore, we have
ð� j �Þ j �0 ¼ ð� j ��0Þ ð3:5Þ
for all �, �0 2 GLþð2,RÞ.Let � be an element of the commensurator ~�� � GLþð2,RÞ of � such that the
associated double coset has a decomposition as in (2.4). If �ðz,X Þ is a Jacobi-likeform in J ð�Þ, we set
TJ ð�Þ�ð Þðz,X Þ ¼Xsi¼1
ð� j �iÞðz,X Þ ð3:6Þ
for all z 2 H. For each � 2 ~��, using (3.5), it can be shown that the power seriesTJ ð�Þ�ðz,X Þ given by (3.6) is an element of J ð�Þ and is independent of the choiceof the coset representatives �1, . . . ,�s. Thus we have a linear endomorphism
TJ ð�Þ : J ð�Þ ! J ð�Þ
of the space J ð�Þ of Jacobi-like forms for � (see [4, Proposition 3.2]).
Definition 3.3 A linear operator TJ ð�Þ : J ð�Þ ! J ð�Þ for � 2 ~�� given by (3.6) iscalled a Hecke operator on J ð�Þ.
Let ð f�Þ1�¼u with u� 0 be a sequence of modular forms with f� 2 M2�ð�Þ for each
� � u, and consider the associated sequence ð��Þ1�¼u of functions �k given by (3.3).
Then by Proposition 3.2 the formal power series �ðz,X Þ ¼P1
k¼u �kðzÞXk is an element
of J ð�Þ.
THEOREM 3.4 If � is an element of ~�� � GLþð2,RÞ having a decomposition as in (2.4),then we have
TJ ð�Þ�ðz,X Þ ¼X1k¼u
Xk�u�¼0
1
�!ð2k� �� 1Þ!
d�
dz�T2k�2�ð�Þfk��ðzÞ� �
Xk,
where T2k�2�ð�Þ is the Hecke operator on the space M2k�2�ð�Þ of modular formsof weight 2k� 2� for � associated to � given by (2.5).
Proof See [4, Theorem 3.5]. g
Jacobi-like forms, differential equations, and Hecke operators 1099
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
4. Linear ordinary differential equations
Let � � SLð2,RÞ be a Fuchsian group of the first kind, and let H� be the union of thePoincare upper half plane H and the set of �-cusps as in section 2. Then the associatedquotient X ¼ �nH� is a compact Riemann surface, which can also be considered as analgebraic curve over C. We denote by K(X ) the function field of X, and choose a non-constant element x of K(X ). Let ’ 2 M1ð�Þ be a meromorphic modular form of weightone for �. Then the functions ’(z) and z’ðzÞ on H can be regarded as functions on Xsatisfying a second order homogeneous linear ordinary differential equation of theform D’,X f ¼ 0 on X with
D’,X ¼d2
dx2þ PX ðxÞ
d
dxþQX ðxÞ ð4:1Þ
that has regular singular points; here PX ðxÞ and QX ðxÞ are elements of K(X ).If f 2 KðX Þ, we have df=dx ¼ ðdf=dzÞðdz=dxÞ and
df
dx¼
df
dz
dz
dx,
d2f
dx2¼
d2f
dz2�
df
dz
d
dzlog
dx
dz
� �� �dz
dx
� �2
,
where z denotes the standard coordinate in C. Using this, we can pull the differentialoperator D’,X in (4.1) back via the natural projection H�!X ¼ �nH�. Thus,the homogeneous equation D’,X f ¼ 0 on X is equivalent to the equation D’ f ¼ 0on H with
D’ ¼d2
dz2þ PðzÞ
d
dzþQðzÞ,
where P(z) and Q(z) are meromorphic functions on H given by
PðzÞ ¼ PXðxðzÞÞdx
dz�
d
dzlog
dx
dz, QðzÞ ¼ QXðxðzÞÞ
dx
dz
� �2
[7, p. 63]. Thus the functions z’ðzÞ and ’(z) for z 2 H are linearly independentsolutions of the associated homogeneous equation D’ f ¼ 0, and the regular singularpoints of D’ coincide with the cusps of � [7].
Given a positive integer m, let SmD’ be the linear ordinary differential operatorof order mþ 1 such that
fzm�i’ðzÞm j 0 � i � mg
is the set of linearly independent solutions of the homogeneous equation SmD’ f ¼ 0.We now consider a more general ordinary differential operator of order n of the form
D ¼dn
dxnþ Pn�1
dn�1
dxn�1þ þ P1
d
dxþ P0,
1100 M. H. Lee
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
where Pi 2 KðX Þ for 0 � i � n� 1. Let S � X be the set of singular pointsof P0, . . . ,Pn�1, and let X0 ¼ X� S. We choose a base point x02X0 and let!1, . . . ,!n be a basis for the space of local solutions of Df ¼ 0 near x0. Then theWronskian
WD ¼ detMD,
is the determinant of the n� n matrix MD ¼ ðdj�1!i=dx
j�1Þ whose (i, j) entry isd j�1
i !i=dxj�1 for 1 � i, j � n. Given x 2 X, let � ¼ f�1, . . . , �n�1g be the set of n� 1
local solutions of Df ¼ 0 near x, and let A� be the ðn� 1Þ � ðn� 1Þ matrix whose(i, j) entry is d j�1�i=dx
j�1 for 1 � i, j � n� 1. Then a function 2 KðX Þ is said tosatisfy the residue conditions with respect to D if the differential ðA� =W Þdx has zeroresidue at every x 2 X0 ¼ X� S for each set � of n� 1 local solutions of Df ¼ 0 near x.
Definition 4.1 An element 2 KðX Þ is said to satisfy the parabolic residueconditions with respect to D if it satisfies the residue conditions and if for each �the differential ðA� =W Þdx has zero residue at every singular point x 2 S wheneverA� is single-valued.
Let � be a positive integer, and let P�,’ be the set of meromorphic functions on H whose associated elements X in K(X ) satisfy the parabolic residue conditionswith respect to S2�D’. Given 2P�, ’, we denote by Sð Þ a solution of the differentialequation S2�D’ f ¼ , and set
��, ’ð Þ ¼d2�þ1
dz2�þ1Sð Þ
’2�
� �: ð4:2Þ
Note that ��, ’ð Þ is independent of the choice of the solution Sð Þ because we have
d2�þ1
dz2�þ11
’2�
X2�i¼0
Ciz2��i’2�
!¼
d2�þ1
dz2�þ1
X2�i¼0
Ciz2�þi
!¼ 0
for any constants Ci 2 C. It can be known that the function ��, ’ð Þ on H given by (4.2)is a meromorphic modular form for � of weight 2�þ 2 [7, p. 32], which leadsto the following lemma.
LEMMA 4.2 The map ��, ’ : P�, ’!M2�þ2ð�Þ given by (4.2) is a one-to-one linear mapof complex vector spaces.
Proof See [3, Lemma 2.3]. g
5. Jacobi-like forms associated to differential equations
Let J ð�Þ be the space of Jacobi-like forms for a Fuchsian group ��SLð2,RÞ ofthe first kind as in section 3, and let
Q1m¼1 Pm, ’ be the space of some meromorphic
functions associated to ’ 2 M1ð�Þ described in section 4. In this section we construct
Jacobi-like forms, differential equations, and Hecke operators 1101
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
a map from J ð�Þ toQ1
m¼1 Pm, ’ and prove that the Hecke operator actions on J ð�Þand on
Q1m¼1 Pm, ’ are compatible with respect to this map.
If w ¼ ð �Þ1�¼1 2
Q1�¼1 P�, ’, we denote by �’ðwÞ the formal power series defined by
�’ðwÞ ¼X1k¼2
Xk�1r¼0
1
r!ð2k� r� 1Þ!
Sð k�r�1Þ
’2k�2r�2
� �ð2k�r�1ÞXk, ð5:1Þ
where Sð k�r�1Þ denotes a solution of the differential equation S2k�2r�2D’ f ¼ k�r�1
as in section 4.
PROPOSITION 5.1 The formula in (5.1) determines a linear map �’ :Q1�¼1 P�, ’! Jð�Þ
of complex vector spaces.
Proof Since clearly �’ is a linear map ofQ1�¼1 P�, ’ into the space R½½X �� of formal
power series, it suffices to show that its image is contained in J ð�Þ. By Lemma 4.2and equation (4.2) we have
��, ’ð �Þ ¼d2�þ1
dz2�þ1Sð �Þ
’2�
� �2 M2�þ2ð�Þ
for all �� 1. Using this and (5.1) we have
�’ðwÞ ¼X1k¼2
Xk�1r¼0
1
r!ð2k� r� 1Þ!ð�k�r�1,’ð k�r�1ÞÞ
ð2k�r�1ÞX k:
Thus the proposition follows from this and Proposition 3.2. g
We now consider another Fuchsian group �0�SLð2,RÞ of the first kind thatis commensurable with � and a nonzero meromorphic modular form ’0 : H! C ofweight one for �0. Given � 2 ~�� with ��� as in (2.4), we can define the Hecke operatorTPð�Þ on
Q1�¼1 P�, ’ by
TPð�Þw ¼ ðTP
� ð�Þð �ÞÞ ð5:2Þ
for each sequence w ¼ ð �Þ1�¼12
Q1�¼1 P�,’, where
ðTP
� ð�Þ �ÞðzÞ ¼ detð�Þ�Xs‘¼1
detð�‘Þ�þ1 �ð�‘zÞ
jð�‘, zÞ2�þ2
’ð�‘zÞ
’0ðzÞ
� �2�þ2 WD’0 ðzÞ
WD’ð�‘zÞ
� �2�þ1
ð5:3Þ
for each �� 1 (see [3, Theorem 4.1]).
1102 M. H. Lee
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
THEOREM 5.2 Given � 2 �, the diagram
Y1�¼1
P�,’�!�’
J ð�Þ
TPð�Þ+ +
TJð�Þ
Y1�¼1
P�, ’0 �!�’0J ð�0Þ
is commutative.
Proof Given w ¼ ð �Þ1�¼1 2
Q1�¼1 P�, ’, by (4.2) and (5.1) we have
�’ðwÞ ¼X1k¼1
�kðzÞXk,
where �1 ¼ 0 and
�k ¼Xk�1r¼0
1
r!ð2k� r� 1Þ!ð�k�r�1, ’ð k�r�1, ’ÞÞ
ðrÞ
for k� 2. Thus, using Theorem 3.4, we obtain
TJ ð�Þð�’ðwÞÞ ¼X1k¼1
Xk�1r¼0
1
r!ð2k� r� 1Þ!
dr
dzrðT2k�2rð�Þ�k�r�1, ’ð k�r�1ÞÞX
k:
However, if TP
� ð�Þ � is as in (5.3), it can be shown that
TP
� ð�Þ � ¼ S2�D’0ðF�, �Þ,
where F�, � is a function satisfying
F�, �
’ 2�0
� �ð2�þ1Þ¼ T2�þ2ð�Þð��, ’ð �ÞÞ
with T2�þ2ð�Þ being the Hecke operator on M2�þ2ð�Þ associated to ��� [3].Hence we obtain
dr
dzrðT2k�2rð�Þ�k�r�1,’ð k�r�1ÞÞ ¼
dr
dzrFk�r�1, �
’ 2k�2r0
� �ð2k�2rþ1Þ¼
Fk�r�1, �
’ 2k�2r0
� �ð2k�rþ1Þ:
Jacobi-like forms, differential equations, and Hecke operators 1103
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014
On the other hand, using (5.1) and (5.2), we have
�’ðTPð�ÞwÞ ¼
X1k¼2
Xk�1r
1
r!ð2k� r� 1Þ!
SðTP
k�r�1ð�Þ k�r�1Þ
’2k�2r�2
!ð2k�r�1ÞX k
¼X1k¼2
Xk�1r¼0
1
r!ð2k� r� 1Þ!�k�r�1 TP
k�r�1ð�Þ k�r�1
ð2k�r�1ÞX k
Hence it follows that �’ TPð�Þ ¼ TPð�Þ �’, and the proof of the theorem
is complete. g
References
[1] Cohen, P., Manin, Y. and Zagier, D., 1997, Automorphic Pseudodifferential Operators, Algebraic aspectsof nonlinear systems (Boston: Birkhauser), pp. 17–47.
[2] Eichler, M. and Zagier, D., 1985, The Theory of Jacobi Forms, Progress in Math., Vol. 55(Boston: Birkhauser).
[3] Lee, M.H., 2001, Pseudodifferential operators associated to linear ordinary differential equations.Illinois Journal of Mathematics, 45, 1377–1388.
[4] Lee, M.H. and Myung, H.C., 2001, Hecke operators on Jacobi-like forms. Canadian MathematicalBulletin, 44, 282–291.
[5] Miyake, T., 1989, Modular Forms (Heidelberg: Springer-Verlag).[6] Miyamoto, M., 2000, A modular invariance on the theta functions defined on vertex operator algebras.
Duke Mathematical Journal, 101, 221–236.[7] Stiller, P., 1984, Special values of Dirichlet series, monodromy, and the periods of automorphic forms.
Memoirs of the American Mathematical Society, 299.
1104 M. H. Lee
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
07:
52 0
5 O
ctob
er 2
014