Hermitian Jacobi Forms of Higher Degree
by
SEAN PATRICK ROBERT HAIGHT
A dissertation accepted and approved in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in Mathematics
Dissertation Committee:
Chair Person, Ellen Eischen
Core Member, Nicholas Addington
Core Member, Alexander Polishchuk
Core Member, Benjamin Young
Representative, Christopher Hendon
University of Oregon
Spring 2024
© 2024 Sean Patrick Robert Haight
This work is openly licensed via CC BY 4.0.
2
DISSERTATION ABSTRACT
Sean Patrick Robert Haight
Doctor of Philosophy in Mathematics
Title: Hermitian Jacobi Forms of Higher Degree
We develop the theory of Hermitian Jacobi forms in degree n > 1. This builds
on the work of Klaus Haverkamp in [Hav95] who developed this theory in degree
n = 1. Haverkamp in turn generalized a monograph of Eichler and Zagier, [EZ85].
Hermitian Jacobi forms are holomorphic functions which appear in certain infinite
series expansions (Fourier Jacobi expansions) of Hermitian modular forms. In this
work we give a definition of Hermitian Jacobi forms in degree n > 1, give their
relationship to more classical Hermitian modular forms and construct a useful tool
for studying Hermitian Jacobi forms, the theta expansion. This theta expansion
allows us to relate our forms to classical modular forms via the Eichler-Zagier map
and thereby bound the dimension of our space of forms. We then go on to apply
the developed theory to prove some non-vanishing results on the Fourier coefficients
of Hermitian modular forms.
3
CURRICULUM VITAE
NAME OF AUTHOR: Sean Patrick Robert Haight
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene
Western Washington University, Bellingham
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2024, University of Oregon
Bachelor of Science, Mathematics, 2018, Western Washington University
AREAS OF SPECIAL INTEREST:
Number Theory
Fourier Analysis
GRANTS, AWARDS, AND HONORS:
Jack and Peggy Borsting Award
Frank W. Anderson Graduate Teaching Award
4
ACKNOWLEDGMENTS
I want to start by thanking my advisor Ellen Eischen. She has helped me understand
more about math and what it’s like to be a mathematician than anyone else in my
life, and for that I am eternally grateful. I am thankful to my committee members,
Nicolas Addington, Alexander Polishchuk, Benjamin Young, and Christopher Hendon
for helping me through the graduation process. For the countless discussions and
many good times I want to thank Corey Brooke, Max Vargas, Bo Phillips, Gautam
Webb, Joe Webster, and Elijah and Holt Bodish. I want to thank Jeff Katen and
Daniel Tarnu for being close companions over the last decade. Finally I want to
thank my family and in particular Rachel Tennant. She has been a deep source of
strength and will for me and a constant reminder that I am not laboring alone.
5
For my Dad.
6
TABLE OF CONTENTS
Chapter Page
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2. Description of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. HERMITIAN JACOBI FORMS . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1. The Hermitian Jacobi Group . . . . . . . . . . . . . . . . . . . . . . . 14
2.2. Hermitian Jacobi Forms . . . . . . . . . . . . . . . . . . . . . . . . . 16
3. HERMITIAN MODULAR FORMS . . . . . . . . . . . . . . . . . . . . . . 18
3.1. Definition and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2. Relationship with Hermitian Jacobi Forms . . . . . . . . . . . . . . . 19
4. THETA EXPANSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1. Definitions and Existence . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2. The Transformation of Theta Functions . . . . . . . . . . . . . . . . . 32
4.3. Linear Independence of Theta Functions . . . . . . . . . . . . . . . . 51
5. THE EICHLER--ZAGIER MAP FOR HERMITIAN JACOBI FORMS . . 54
5.1. Eichler--Zagier Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2. Twists of the Eichler--Zagier Map . . . . . . . . . . . . . . . . . . . . 62
6. NON-VANISHING FOURIER COEFFICIENTS . . . . . . . . . . . . . . . 67
6.1. Vector-valued Hermitian Modular Forms . . . . . . . . . . . . . . . . 67
6.2. Vector-Valued Hermitian Jacobi Forms . . . . . . . . . . . . . . . . . 71
6.3. Non-zero Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . 76
7
7. FUNDAMENTAL FOURIER COEFFICIENTS . . . . . . . . . . . . . . . 81
7.1. Difficulties and Roadblocks . . . . . . . . . . . . . . . . . . . . . . . . 81
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8
CHAPTER 1
INTRODUCTION
We develop core aspects of the theory of Hermitian Jacobi forms in degree
n > 1. This builds on the work of Klaus Haverkamp in [Hav95] who developed
this theory in degree n = 1. Haverkamp was in turn generalizing a monograph
of Eichler and Zagier [EZ85]. Hermitian Jacobi forms are holomorphic functions
which appear in certain infinite series expansions (Fourier Jacobi expansions) of
Hermitian modular forms. In this work we give a definition of Hermitian Jacobi
forms in a higher-dimensional setting, give their relationship to more classical Hermitian
modular forms and construct a useful tool for studying Hermitian Jacobi forms,
the theta expansion. This theta expansion allows us to relate our forms to classical
modular forms via the Eichler--Zagier map and thereby bound the dimension of our
space of forms. We then apply the developed theory to prove some non-vanishing
results on the Fourier coefficients of Hermitian modular forms.
1.1 Motivation
Broadly speaking, the goal of this paper is add an additional tool to studying
a class of functions known as automorphic forms. These functions can generally be
thought of as holomorphic functions on some complex manifold, sometimes identified
with the moduli space of some family of geometric objects, that satisfy symmetry
properties with respect to a linear algebraic group that acts on the domain manifold.
Auotomorphic forms have broad applications in number theory, for example the
application of classical modular forms in Andrew Wiles’ solution to Fermat’s Last
Theorem. Today many mathematicians study automorphic forms as objects of
interest in their own right or to help shed light on a very broad family of conjectures
9
in the field known as the Langlands Program.
The study of automorphic forms can be divided into different subfields based
on the group that governs the symmetry of these forms. Classically this group was
SL2(Z) with which we associate classical modular forms. In his 1939 paper [Sie39]
Carl Ludwig Siegel introduced a higher dimensional analogue known as Siegel modular
forms which transform with respect to the group Spn(Z). The class of automorphic
forms that we are most interested in here are those known as Hermitian modular
forms introduced by Hel Braun in three papers, [Bra49], [Bra51a], and [Bra51b] .
We give a formal definition and some introduction to the theory in chapter 3.
In [EZ85], Eichler and Zagier formally introduced Jacobi forms, a class of C-valued
functions on H × C that satisfied some symmetry conditions with respect to the
group SL2(Z) as well as some holomorphy conditions. Two important examples
include Jacobi’s theta function and the Weierstrass ℘ function which parametrizes
the complex points of an associated elliptic curve. Jacobi forms also appear as
the coefficients in a certain series expansion of Siegel modular forms known as the
Fourier Jacobi expansion. In this way Jacobi forms have proven to be a useful tool
in studying Siegel modular forms.
In [Hav95] Haverkamp gives a generalization of the theory of Jacobi forms to the
setting of Hermitian modular forms, though only in degree 2. This paper is the
most direct logical predecessor to the current work. Even with this limit on the
degree, there are applications of this work, principally to the study of the Fourier
coefficients of Hermitian modular forms of degree 2, as in [AD19]. Our goal in this
work is to generalize Haverkamp’s work to an arbitrary degree n. As an initial
motivation we hoped to extend the results of [AD19] and generalize similar results
of [BD22]. As often happens, we found an insurmountable difficulty that is discussed
in the final chapter.
10
1.2 Description of results
Our primary result is the following. Let K be a quadratic imaginary field with
ring of integers O.
Theorem 1.1. Let ϕ be an Hermitian Jacobi form of weight k, degree n ≥ 1 and
invertible index T . Then ϕ has a theta∑expansion of the form
ϕ(τ, w, z) = hs(τ)θT,s(τ, w, z) (1.1)
n
s∈(O#) /TOn
where each hs is a classical modular form.
Here w, z ∈ Cn and τ ∈ H the complex upper-half plane. We define θT,s in
Equation (4.7). This is Theorem 5.5 in the main text. In Chapter 2 we give new
definitions of Hermitian Jacobi forms in degree n > 1. These directly generalize
those of Haverkamp. These definitions are chosen so that in Chapter 3 we can prove
the following proposition.
Proposition 1.2. Let F be an Hermitian modular form. Then F has a Fourier
Jacobi expansion


τ w ∑F = ϕT (τ, w, z)e(TZ) (1.2)
z T T∈Λn(O)
where each ϕT is an Hermitian Jacobi form.
Here { }
Λn(O
t
K) := A ∈ M #n(C)|A = A , ai,i ∈ Z, ai,j ∈ O (1.3)
and e(TZ) is shorthand for e2πitr(TZ). This is Proposition 4.1 in the main text. We
give an introduction to the theory of Hermitian modular forms in Chapter 3 before
this proposition.
11
Chapter 4 is the technical heart of the paper. In this chapter we first introduce our
theta functions and explain the existence of the expansion given in Theorem 5.5.
The bulk of this chapter is establishing Proposition 4.8, a transformation law for
these theta functions under an action of a congruence subgroup of SL2(Z). The fact
that each hs in the theta expansion is a modular form will eventually follow from
this transformation law. In chapter 5 we begin by translating the transformation
law for the theta functions into transformations of the coefficients. After proving
the theorem above we move onto generalizing the Eichler--Zagier map and the twists
there-of to our setting. These functions, which take Hermitian Jacobi forms to
classical modular forms, are a central tool in [AD19], Anamby and Das’s paper on
non-vanishing fundamental Fourier coefficients. In chapter 6 we introduce some of
the theory of vector valued Hermitian modular forms in order to give some novel
applications of our work to the theory of Hermitian modular forms. In this chapter
we prove the following result.
Proposition 1.3. Let F be a non-zero, possibly vector valued, Hermitian modular
form of degree n ≥ 2. Then F has infinitely many non-zero Fourier Jacobi coefficients
with non-singular index.
The above result essentially tells us that each Hermitian modular form gives
rise to many Hermitian Jacobi forms. Using this and the theta expansion we prove
the proposition below.
Proposition 1.4. Let F be a non-zero Hermitian modular form of weight k and
degree n ≥ 2. Then F has infinitely many non-zero Fourier coefficients with index
12
 m rtA =  such that
r T
det(A) ≤ C 2k,n−1,D det(T ) (1.4)
where
n
(k − n)D3+2⌊ ⌋2
Ck,n,D = (1.5)
12
and D is |∆K |, the discriminant of the number field K.
In the final section we explain the relationship between this work and the existence
of non-zero fundamental Fourier coefficients. As stated in the motivation the initial
goal of this work was to generalize [BD22, Theorem 1.1] of Böcherer and Das, stated
below, to the Hermitian setting.
Theorem 1.5 (Theorem 1.1 of [BD22]). Let F be a non-zero vector valued Siegel
modular form of weight ρ and degree n. Suppose further that k(ρ)− n ≥ ϱ(n). When
2
n is even, assume that F is cuspidal. Then there exists infinitely many GLn(Z)
inequivalent matrices T ∈ Λ+n such that dT is odd and square free, and aF (T ) ̸= 0.
Anamby and Das have proven a similar result, [AD19, Theorem 1], for Hermitian
modular forms when the degree n = 2 and heavily leveraged the theory of Hermitian
Jacobi forms in this setting. While we were not able to generalize the approaches of
either of these two papers here, we do outline the general strategy and where the
road blocks are to generalizing these approaches.
13
CHAPTER 2
HERMITIAN JACOBI FORMS
2.1 The Hermitian Jacobi Group
The goal of this chapters is to give the definition of Hermitian Jacobi forms in
degree greater than or equal to 1. Let n ≥ 1 be a integer and K be a quadratic
imaginary field with ring of integer O. Let D = |∆K | be the absolute value of
the discriminant of K. Hermitian Jacobi forms of degree n are functions ϕ : H ×
Cn × Cn → C which satisfy some transformation properties and admit a Fourier
expansion. Before we can define Hermitian Jacobi forms, we need to define a group
action on H× Cn × Cn. Let
{ }
t
 U(n, n) := A ∈ M2n(O) : A JnA = Jn (2.1)
 0 −Inwhere Jn = . The group U(n, n) is the unitary group of degree n which
In 0
governs the transformations of Hermitian modular forms of degree n. We define a
right action of U(1, 1) on Cn × Cn via
[λ, µ] · A := [λ, µ]A = [a11λ+ a21µ, a12λ+ a22µ] (2.2)
for any λ, µ ∈ Cn × Cn and A = [aij] ∈ U(1, 1). This action gives rise to the
following definition.
Definition 2.1. Define multiplication in Γ1 := U(1, 1)⋉C2n via
(A,X1) · (A,X2) := (A1A2, X1 · A2 +X2). (2.3)
This group will be that which governs the transformation law of our Hermitian
Jacobi forms.
14
Let (τ, w, z) ∈ H × Cn × Cn. Throughout the paper we consider w to be a row
vector and z to be a column vector. In order to define our group action of Γ1 on
H × Cn × Cn we define actions of the two constituent groups. First for [λ, µ] ∈
Cn × Cn, where both λ and µ are row vectors, we define
t
[λ, µ] · (τ, w, z) := (τ, w + λτ + µ, z + λ τ + µt). (2.4)
Here we write At for the transpose of a matrix. Often in related literature the convention
is tA. Next we consider the action of U(1, 1) on (τ, w, z). We also note that
  U(1, 1) = {ϵM : ϵ ∈ O×,M ∈ SL2(Z)}. (2.5)a bFor M =  ∈ SL (Z) define Mτ = aτ+b2 and j(M, τ) = cτ + d. We definecτ+d
c d
an action of U(1, 1) on H× Cn × Cn b(y )
ϵM · ϵw ϵz(τ, w, z) := Mτ, , . (2.6)
j(M, τ) j(M, τ)
We then define a group action of Γ on H× Cn × Cn1 via
(A, [λ, µ]) · (τ, w, z) := A · ([λ, µ] · (τ, w, z)). (2.7)
Remark 2.2. One can check that this is a well-defined group action by first computing
that, for A ∈ U(1, 1) and [λ, µ] ∈ Cn × Cn,
[λ, µ] · (A(τ, w, z)) = A([λ, µ] · A) · (τ, w, z). (2.8)
With the group actions defined above we can define associated slash operators.
Let ϕ : H × Cn × Cn → C be a function. For ϵM ∈ U(1, 1), λ, µ ∈ Cn, an integer k
and T ∈ Λn(O) define
ϕ|T,kϵM(τ, w, z) := ϵ−k(j(M, τ))−ke (−cw(Tz/j(M, τ))ϕ (ϵM · (τ, w, z))) (2.9)
| t t tϕ T (λ, µ)(τ, w, z) := e(λTλ τ + wTλ + λTz)ϕ τ, w + λτ + µ, z + λ τ + µt .
(2.10)
15
2.2 Hermitian Jacobi Forms
In the previous section we introduced the transformation group for our Hermitian
Jacobi forms. The other condition on our forms is the existence of a Fourier expansion
of a certain form. In order to give this condition explicitly we introduce a couple
indexing sets for these expansions. Let O# := √i O be the different of our ring of
D
integers.
Remark 2.3. The set O# is the dual lattice of O with respect to the trace pairing
on K. In particular, for each α ∈ O# and β ∈ O we have
2Re[αβ] ∈ Z. (2.11)
Define { }
t
Λn(O) := A ∈ Mn(C)|A = A , ai,i ∈ Z, a #i,j ∈ O (2.12)
and
Λ+n (O) := {A ∈ Λn(O) : A is positive definite } . (2.13)
Finally define our indexing set, for a matrix T ∈ Λn(O), ( )   n m rt S T := (m, r) ∈ Z× O# :   ∈ Λ+n+1(O) . (2.14)r T
Now we can define an Hermitian Jacobi form.
Definition 2.4. Let k be an integer and T ∈ Λn(O). A holomorphic function ϕ :
H×Cn×Cn → C is an Hermitian Jacobi form of weight k, degree n, and index T if
1. ϕ satisfies the following transformation laws
ϕ|T,kA(τ, w, z) = ϕ(τ, w, z) for all A ∈ U(1, 1) (2.15)
ϕ|T [λ, µ](τ, w, z) = ϕ(τ, w, z) for all [λ, µ] ∈ On ×On (2.16)
16
2. ϕ has a Fourier expansion
∑
ϕ(τ, w, z) = α(m, r)e(mτ + wr + rtz). (2.17)
(m,r)∈ST
We call this space J nT,k(O) or just J nT,k if O is clear from context.
These transformations given in condition (1) above can be rephrased as a single
requirement using the group Γ1 though we find it easier to think of these two transformations
separately.
17
CHAPTER 3
HERMITIAN MODULAR FORMS
3.1 Definition and Notation
Hermitian modular forms are a higher dimensional analogue of classical modular
forms, similar to Seigel modular forms. Whereas Seigel modular forms transform
with respect to the symplectic group, Hermitian modular forms transform with
respect to the unitary group U(n, n) for some integer n ≥ 1. These objects were
first introduced in a sequence of three papers by Hel Braun [Bra49], [Bra51a] and
[Bra51b]. Since their definition, these forms have grown to be an important class
of automorphic forms with a theory that parallels many of the developments in the
theory of Seigel modular forms. In this chapter we give an overview of this theory
and explain the relationship between Hermitian modular forms and Hermitian Jacobi
forms.
First define the Hermitian upper-half space of degree n by
Hn := {Z ∈ Mn(C) : (Z −
t
Z )/2i > 0}. (3.1)
The group U(n, n) has an action on Hn defined by
  M⟨Z⟩ := (AZ +B)(CZ +D)
−1 (3.2)
for A BM =  ∈ U(n, n) and Z ∈ Hn. Now we define Hermitian modular forms.
C D
Definition 3.1. Let k be an integer. A function F : Hn → C is a Hermitian
modular form of weight k, and degree n for the group U(n, n) if it is holomorphic
18
and, for any 
 
A BM =  ∈ U(n, n) and Z ∈ Hn, we have
C D
F (Z) = F |kM(Z) := det(CZ +D)−kF (M⟨Z⟩). (3.3)
Here A,B,C and D are n × n block matrices. When n = 1 we require that F be
holomorphic at the cusps of SL2(Z) as well. Concretely this means that we require
that F (Z) is bounded as Im(z) tends to infinity. We denote the space of such functions
by Mnk(O).
For a matrix B define e(B) = e2πitrB. Then a Hermitian modular form F has a
Fourier expansion ∑
F (Z) = a(F,A)e(AZ). (3.4)
A∈Λn(O)
If instead the above sum is over Λ+n (O) we say that F is a cusp form.
3.2 Relationship with Hermitian Jacobi Forms
We now explain the relationship between Hermitian Jacobi forms and Hermitian
modular forms. The basic principle is that the Fourier series can be rearranged in
order to give F as a sum of Hermitian Jacobi forms multiplied by an exponential
term. We first need a lemma in order to verify the transformation law of these
coefficients.
Lemma 3.2. Let µ, λ ∈ On. Then for any T ∈ Λn(O) we have that
λTµt
t
+ µTλ ∈ Z. (3.5)
Proof. We have, using(the formula fo)r th∑e tr∑ace of a matrix product,n n
tr t tµ Tλ+ λ Tµ = tijµjλi + tjiλjµi. (3.6)
i=1 j=1
19
The terms for which i = j are in Z(since )
tii ∈ Z and µjλi + λjµi ∈ O ∩ R = Z. (3.7)
Then we have ∑n ∑n
tr t(µtTλ+ λ Tµ) ≡ 2Re[tijµiλj] (mod Z) (3.8)
i=1 j=1
j≠ i
Since tij ∈ O# and µiλj ∈ O, Remark 2.3 implies each term in the above sum is an
integer. Thus tr t(µtTλ+ λ Tµ) is as well.
With this we can prove the that the Fourier Jacobi coefficients of an Hermitian
modular form are Hermitian Jacobi forms.
Proposition 3.3. Let F : Hn+1 → C be a Hermitian modular form of weight k. Let
F have Fourier expansion ∑
F (Z) = a(F,A)e(AZ).
A∈Λn+1(O)
For T ∈ Λn(O) define
∑   m rt
ϕT (τ, w, z) := aF,  e(aτ + wr + rtz).
(m,r)∈ST r T
Then 

τ w ∑F = ϕT (τ, w, z)e(TZ) (3.9)
z Z T∈Λn(O)
and each ϕT is an Hermitian Jacobi form of degree n, weight k and index T .
Proof. First we address the holomorphy concerns of ϕT . The Fourier series of F
converges absolutely on compact subsets of thedomain Hn+1. H∑   
ence the sub-sum
 m rte(TZ)ϕT (τ, z, w) = a F,  em rtτ w (3.10)
(m,r)∈ST r T r T z Z
20
will converge absolutely on compact subsets of Hn and thus so too will the sum
defining ϕT . In particular we see that ϕT is a well-defined holomorphic function on
H× Cn × Cn with the proper Fourier expansion.
In order to prove that ϕT satisfies the necessary transformation laws we first
need Equation (3.9). If  m rt ∈ Λn+1(O) (3.11)
r T
then T ∈ Λn(O). Observe that  m rt τ w
e(TZ)e(mτ + wr + rtz) = e   . (3.12)
r T z Z
so that Equation (3.9) immediately follows from the definition of ϕT .
It remainsto check that ϕT satisfies the proper transformation law. Let ϵ ∈
a b
O× and  M =   ∈ SL2(Z). Note that
c d
 ϵa 0 ϵb 0 
M :=

0 In 0 0
ϵc 0 ϵd 0 

 ∈ U(n+ 1, n+ 1). (3.13)
0 0 0 In
Since F is an Hermitian modular form of weight k, we have
F |kM = F. (3.14)
If follows from direct com〈putation that 〉  τ w  Mτ ϵwM j(M,τ) =  . (3.15)
z Z ϵz Z − czw
j(M,τ) (cτ+d)
21
Expanding Equation (3.14) using Equation (3.9) and comparing the coefficients of
e(TZ) gives Equation (2.15) for ϕT .
Next consider our other transformation requirement. Let λ, µ ∈ On. Again, we
consider these as row vectors. To get Equation (2.16) for ϕT we use the matrix 1 0 0 µ

t t 
λ I tn µ λ µ
N :=
 0 0 1 −λ
 . (3.16)
0 0 0 In
We hav〈e  〉
 
τ w  τ λτ + w + µN =  . (3.17)
t t t t
z Z λ τ + µt + z Z + λ w + zλ+ τλ λ+ (λ µ+ µtλ)
If we expand Equation (3.14) using Equation (3.9) and compare the coefficients we
almost get the desired equality. On the left-hand side we have an extra term of
t
e(λTµt + µTλ ) = 1
by Lemma 3.2. Equation (2.16) follows.
The expansion above is known as the Fourier Jacobi expansion of an Hermitian
modular form. By studying the Fourier expansion of Hermitian Jacobi forms, we
can get information about Hermitian modular forms, similar to the Siegel setting,
where the theory of Jacobi forms sheds light on Siegel modular forms.
22
CHAPTER 4
THETA EXPANSIONS
This chapter is the most technically dense. We introduce our theta functions,
relate them to Hermitian Jacobi forms and study their transformations. We’ll almost
exclusively be studying Hermitian Jacobi forms of positive definite index as non-
singularity of the index is required for the definition of the theta functions. Proposition
6.11 tells us this requirement is not too restrictive.
4.1 Definitions and Existence
Throughout this section let ϕ be a Hermitian Jacobi form of degree n, weight k
and index T ∈ Λ+n (O) with Fourie∑r expansion
ϕ(τ, w, z) = α(m, r)e(mτ + wr + rtz). (4.1)
(m,r)∈ST
Since T ∈ Λ+n (O) we see that T is necessarily non-singular. We introduce some
notation that will be helpful for this section.
Define, for A ∈ Λn(O)
 ( √ )|∣∣det i (DA | )∣∣ if n is evend(A) := ∣ √ . (4.2)√−i det i DA ∣ if n is odd
D
This quantity is called the content ofthe matrix A. For n ≥ 1 and D ∈ R letD if n is oddEn(D) =  . (4.3)1 if n is even
This is a correction term to d(A) that is necessary to relate the content of an n × n
matrix to the content of a{n n+ 1× n+ 1 matrix. Finally let }
N
IT,s := N ∈ Z>0 : + stT−1s ∈ Z . (4.4)
En(D)d(T )
23
This will be the indexing set of our theta coefficients. The main result of this sections
is
( )
Proposition 4.1. Let ϕ be as above. For s ∈ O# n and an integer N > 0 define
αs+TOn(N 
)= a(m, r) (4.5)
m rtif r ≡ s (mod TOn) and N = d . Define for τ ∈ H and w, z ∈ Cn,
r T
∑ ( )N
hs(τ) := αs+TOn(N)e τ (4.6)
En(D)d(T )
N∈IT,s∑ ( − )θ t 1 tT,s(τ, w, z) := e r T rτ + wr + r z . (4.7)
n
r∈(O#)
r≡s (mod TOn)
Then we have
∑
ϕ(τ, w, z) = hs(τ)θT,s(τ, w, z) (4.8)
n
s∈(O#) /TOn
We will see that the quantity En(D)d(T ) will be the level of the theta coefficients
hs of our Hermitian Jacobi form ϕ. In order to prove Proposition 4.1 we need 4
preliminary results. We first argue that our theta functions are well-defined holomorphic
functions.
Proposition 4.2. Let s ∈ (O#)n. Then
∑ ( )
e rtT−1rτ + wr + rtz (4.9)
r∈(O#)n
r≡s (mod TOn)
is absolutely and uniformly convergent on compact subsets of H× Cn × Cn.
Proof. Let K ⊂ H× Cn × Cn be compact.
24
First we find ω1, ω2, ω3 > 0∣su(ch that∣e rtT− )∣ ( )1rτ ∣ ≤ exp −ω1||r||2 (4.10)
∣|e((wr))∣| ≤ exp (ω2||r||) (4.11)∣e rtz ∣ ≤ exp (ω3||r||) (4.12)
for all (τ, w, z) ∈ K and r ∈ (O#)n with ||r|| sufficiently large. If d > 0 is the
minimal eigenvalue of T−1 then
|rtT−1r| ≥ d||r||2. (4.13)
This follows from the fact that, because T−1 is a positive definite, Hermitian matrix,
T−1 is orthogonally diagonalizable.
Since K ⊂ H × Cn × Cn is compact we can choose some v0 > 0 so that Im(τ) > v0
for all (τ, w, z) ∈ K. We∣have∣ (e rtT− )∣ ( )1rτ ∣ ≤ exp −2πdv0||r||2 . (4.14)
Let ω1 = 2πdv0. Inequalities 4.11 and 4.12 just follow from the Cauchy--Schwarz
inequality and the fact that w and z are such that (τ, w, z) ∈ K for some τ .
These togethe∣r give∣ (e rtT− )∣ ( )1rτ + wr + rtz ∣ ≤ exp −ω 21||r|| + (ω2 + ω3)||r|| (4.15)
for r with ||r|| sufficiently large. If 0 ≤ ω4 < ω1 then for ||r|| sufficiently large we
have
−ω 24||r|| > −ω1||r||2 + (ω2 + ω3)||r||. ( ) (4.16)
Let Mr = exp
n
(−ω ||r||24 ). The for all (τ, w, z) ∈ H × Cn × Cn and r ∈ O# with
||r|| sufficiently large we ha∣ve∣ (e rtT− )∣1rτ + wr + rtz ∣ ≤ Mr. (4.17)
25
By the Weierstrass M -test it suffices to show that
∑
Mr < ∞. (4.18)
r∈(O#)n
r≡s (mod TOn)
Let Λ = {r ∈ (O#)n : r ≡ s (mod TOn)}. We see that Λ is a shift of a lattice in Cn
so that, for
ΛN = {λ ∈ Λ : (N − 1) ≤ ||λ|| ≤ N} (4.19)
we have |ΛN | ≤ CN2n for some fixed constant C and N sufficiently large. After
splitting sum (4.18) into a sum over N and a sum over ΛN we see convergence
follows from the ratio test.
In order to ensure that the coefficients αs+TOn(N) are well-defined, we need
the following lemma.
Lemma 4.3. Let r,m, s and p besuchthat
m rt 

, p st ∈ Λn+1(O). (4.20)
r T s T
If s ≡ r (mod TOn) and    m r p sdet  = det 
rt T st T
then a(m, r) = a(p, s)
Proof. There are two useful equations for this proof. Applying Equation (2.16)
with µ = 0 and comparing coefficients of e[mτ + wr + rt] in the Fourier expansion
of ϕ gives
t t t
a(m, r) = a(m+ λTλ + rλ+ rtλ , r + Tλ ) (4.21)
26
for each λ ∈ On.
Since T is non-singular we can write
 m rtdet  = det(T )(m− rtT−1r) (4.22)
r T
If
det(T )(m− rT−1rt) = det(T )(p− sT−1st) (4.23)
then
m− rT−1rt = p− sT−1st. (4.24)
Let tλ = T−1(s− r) ∈ On. We then compute
rt
t t
λ + λr + λTλ = stT−1s− rtT−1r. (4.25)
Thus
a(m, r) = a(m+ sT−1st − rT−1rt, s) = a(p, s)
as desired.
We need two more lemmas in order to prove Proposition 4.1. Both are of a
similar flavor and are used in comparing the Fourier expansion of ϕ with the theta
expansion.
Lemma 4.4. Let M ∈ Λ+n (O). Then d(M) ∈ Z>0.
Proof. First consider the case when n i∣s even. Then∣∣ √ ∣∣d(M) = det(i DM)∣ . (4.26)
27
( √ ) √
We have det i DM ∈ O because i DM ∈ Mn(O). Since M ∈ Λ+n (O) we have
t
M = M .(Thus√ ) √ √ √
det (i DM )= det(−i DM) = (−1)
ndet(i DM) = det(i DM). (4.27)
√ √
Thus det i DM ∈ R ∩ O = Z and since M is positive definite det(i DM) ≠ 0
and d(M) ∈ Z>0
Now consider when n is odd. In this∣case we have∣∣∣ −i √ ∣∣d(M) = √ det(i DM)∣∣ . (4.28)D
√ √
As in the preceding argument, √−i det(i DM) ∈ R. We argue that √−i det(i DM) ∈
D D
√
O. Let A = i DM . The important characteristics of A for the following are that
t √
A = −A , aij ∈ O and aii ∈ i DO. Let Sn denote the symmetric group on n
letters. We have ∑ ∏
det(A) = (−1)σ ai,σ(i). (4.29)
∐ σ∈Sn
Write Sn = P Q where P∑= {σ ∈ S : σ = σ
−1
∏n ∑} and Q =∏Sn\P . Then
det(A) = (−1)σ ai,σ(i) + (−1)σ ai,σ(i). (4.30)
σ∈P σ∈Q
For σ ∈ Q we have σ ̸= σ−1 so we can pair summands of index σ with summands of
index σ−1. We have∏ ∏ ∏
− σ − σ−1 − − σ−1( 1) ai,σ(i) = ( 1) ai,σ(i) = ( 1) ∏ aσ(i),i (4.31)
− − σ−1( 1) ai,σ−1(i). (4.32)
The terms in the sum over Q after being grouped in this way will be of the form
2iIm(a) for some a ∈ O. We have ( )
√−i −i2iIm(a) = 2Re √ a ∈ Z. (4.33)
D D
28
Now consider the sum over P . If σ is written as a product of disjoint cycles then
the order of σ will be the length of the largest cycle. We see that σ must be a product
of disjoint transpositions because σ has order two. Because n is odd σ must have a
√
fixed point. If σ(i) = i then ai,σ(i) = ai,i ∈ i DO. Hence∑ ∏ √
det(A) = (−1)σ ai,σ(i) ∈ i DO (4.34)
σ∈P
√
so √−i det(i DM) ∈ O ∩ R = Z. Thus d(M) ∈ Z>0 when n is odd.D
Lemma 4.5. Let s ∈ (O#)n. Then
En(D)d(T )s
tT−1s ∈ Z. (4.35)
Proof. First consider when n is odd. In this case we have
En(D) = D (4.36)
i √
d(T ) = ±√ det(i DT ). (4.37)
D
We see
√ √ √
d(T )T−1 = ± det(i DT )(i DT−1) = Adj(i DT ) ∈ Mn(O). (4.38)
Then ( √ )t √ √
En(D)d(T )sT
−1s = ± i Ds Adj(i DT )(i Ds) ∈ O. (4.39)
We also have En(D)d(T )stT−1s ∈ R because T is Hermitian so En(D)d(T )stT−1s ∈
Z as desired.
Now consider when n is even. In this case we have
En(D) = 1 (4.40)
√
d(T ) = ± det(i DT ). (4.41)
29
√ √
Note that the diagonal entries of Adj(i DT ) are the determinants of form det(i DA)
for some A ∈ Λn−1(O). Since n − 1 is odd we see that these diagonal entries lie in
√
i DO, as in Equation (4.34). Thus
√ √
d(T )T−1 = ±i DAdj(i DT ) (4.42)
√
is Hermitian with diagonal entries in DO and off diagonal entries in i DO. We
have
∑n ∑
sAst = a 2ii|si| + 2Re[aijsisj]. (4.43)
i=1 i<j
√
Since aii ∈ DO aii|s |2i ∈ Z. Since aij ∈ i DO we have aijsi ∈ O. Thus 2Re[aijsisj] ∈
Z because sj ∈ O#. Thus d(T )sT−1st ∈ Z as desired.
We now prove the existence of theta expansions for Hermitian-Jacobi forms.
Proof of Proposition 4.1. We first recall the setting. We have an Hermitian Jacobi
form ϕ with Fourier expansion
∑
ϕ(τ, w, z) = α(m, r)e(mτ + rw + rtz). (4.44)
(m,r)∈ST
Furthermore we have for τ ∈ H and w, z ∈ Cn,
∑ ( )N
hs(τ) := αs+TOn(N)e τ (4.45)
∑ ( En(D)d(T )N∈IT,s )
θ (τ, w, z) := e rtT−1rτ + wr + rtT,s z (4.46)
n
r∈(O#)
r≡s (mod TOn)
where { }
Is,T := N ∈
N
Z : + stT−1≥0 s ∈ Z (4.47)
En(D)d(T )
30
and
αs+TOn(N)= a(m, r) (4.48)
m rt
if r ≡ s (mod TOn) and  N = d .
r T
First we argue that 
   
N = dm rt m rtfor some   ∈ Λn+1(O) with
r T r T  
m rt
r ≡ s (mod TOn) if and only if N ∈ IT,s. First suppose  N = d . The
r T
fact that N ∈ Z>0 is the content of Lemma 4.4.
Consider N + stT−1s. We have
En(D)d(T )
N
+ stT−1s = m− rtT−1r + stT−1s (4.49)
En(D)d(T )
by Equation (4.22). An application of Lemma 3.2 shows that if r ≡ s (mod TOn)
stT−1s− rtT−1r ∈ Z. (4.50)
We then see that N ∈ IT,s.
Next suppose N ∈ IT,s then take m =m(N, r) := N + rtT−1r for anyEn(D)d(T ) 
m rt m rt
r ≡ s (mod TOn) so that  N = d  . To see that   > 0 use that,
  r T r T
for x1 ∈ C× Cn we have
x
[ ]  m rtx1 ( )| |2 N tx xt1 = x1 + (x+ T−1x −11r) T (x+ T x1r) > 0
r T x En(D)d(T )
(4.51)
31
 x1as long as  ≠ 0. To get
x
∑
θshs = ϕ(τ, w, z) (4.52)
s∈(O#)n/TO
write out both sums and use the argument above to realize (s, r,N) 7→ (m(N, r), r)
gives a bijection between the two indexing sets.
We will see that the functions hs are classical modular forms and, since the
Fourier coefficients of hs are directly related to those of ϕT (and hence to F ), we
can use our knowledge of classical modular forms to gain information about Hermitian
modular forms. We explain this process in more detail in Chapter 6.
4.2 The Transformation of Theta Functions
In order to prove that the hs are modular forms we will prove some transformation
laws for our theta functions and then use the transformation law satisfied by our
Hermitian Jacobi forms to translate these into transformation laws satisfied by the
hs. We introduce some notation.
We then define a character χD : Z → C(by)the Kronecker symbol
D
χD(q) := . (4.53)
q
This is a character modulo D because D is a fundamental discriminant. Recall the
group
  
Γ0(r) := a b

∈ SL2(Z) : c ≡ 0 (mod r) . (4.54)c d
The primary result of this section is as follows:
32
 
a b
Proposition 4.6. For the theta function  θT,s and M =   ∈ Γ0 (En(D)d(T ))
c d
we have
1 ∑ [ ( )]
θ nT,s|T,nMJ = √ χD(d) e a 2Re[stT−1s′] + bstT−1s θT,s′ .
det(−i DT )
′ ns ∈(O#) /TOn
(4.55)
We follow Haverkamp [Hav95, Satz 4.5] closely for this proof. The steps are as
follows.
1. Introduce auxiliary functions ψ and ϕ and prove these satisfy two useful transformations
by computing some related exponential integrals.
2. Relate these two functions to our theta functions using Fourier expansions.
3. Translate the transformation laws of ψ and ϕ into those of the theta functions.
4. Relate a Gauss-like sum that appears to our character χD.
First we’ll introduce these auxiliary functions and prove two transformation laws
for ψ and ϕ.  
a b
Definition 4.7. Let s, σ, w, z ∈ Cn,   ∈ Γ0(En(D)d(T )) and τ ∈ H. Again,
c d
we consider w as a row vector and s, σ and z as column vectors. Define
ϕ(w, z) = e[wT∑−1z(aτ +[b)a] ] (4.56)
ψs,σ(w, z) = e r
tTrτ + (wr + rtz)(aτ + b) + 2Re[σtr] . (4.57)
r≡s (mod On)
Similarly to Proposition 4.2 ψs,σ is given by a series that converges absolutely and
uniformly on compact subsets of Cn × Cn.
33
Remark 4.8. We assume that a ̸= 0. If a = 0 then c = ±1 so En(D)d(T ) = 1. We
see En(D) = 1 so that n is even and Γ0(En(D)d(T )) = SL2(Z). If we prove for any
M ∈ SL2(Z) with a ̸= 0 then we see that for any M ∈ SL2(Z) with a ≠ 0 we have
  θT,s|T,nMJ = θT,s. (4.58)
Since 1 1 J−1 and JJ−1 have non-zero (1,1) entry we see that θT,s|T,nA = θT,s
0 1
for every A ∈ SL2(Z). In particular we have this equation for MJ when a = 0.
Thus we can assume a ̸= 0 from here on out.
Now we prove the transformation laws for ϕ and ψ.
Lemma 4.9. Let α ∈ TOn and β ∈ (O#)n. Then we have
[ ]
ϕ(w + αt, z + α)ψs,σ(w + α
t, z + α) = e 2Re[bstα− aσtT−1α] ϕ(w, z)ψs,σ(w, z)
(4.59)
and
t t
ϕ(w + β (JMτ), z + β(JMτ))ψ(w + β (JMτ), z + β(JMτ)) =
[ ( )] (4.60)
t t
e − β T−1βJMτ + wT−1β + β T−1z + 2Re((dstβ − cσtT−1β)) ϕ(w, z)ψ(w, z).
(4.61)
Proof. First we consider the proof of Equation (4.59). We have
ϕ(w, z)ψ∑s,σ(w, z)[ (]4.62)
= e wT−1z(aτ + b)a+ rtTrτ + (wr + rtz)(aτ + b) + 2Re[σtr] .
r≡s (mod On)
(4.63)
34
Note that we can replace r in the above sum by r + aT−1α and it will leave the
whole sum unchanged because aT−1α ∈ On. If we add and subtract b(αtr + rtα +
aαtT−1α) within our exponential then our rth term will be
[ ]
e w[T−1z(aτ + b)a+ αtT−1α(aτ + b)a+ (wT−1α + αtT−1z)(a)(aτ + b)] (4.64)
·e rtTrτ + (wr + rtz)(aτ + b) + rtα(aτ + b) + αtr(aτ + b) + 2Re[σtr] (4.65)
·e[2Re[σt(aT−1α)− brtα]] · e[−abαtT−1α]. (4.66)
Since α ∈ TOn we have αtT−1α ∈ Z so this last term is just 1. Since r ≡ s
(mod On) we have
e[2Re[brtα]] = e[2Re[bstα]]. (4.67)
Hence our sum for ϕ · ψ has rth term
[ ] [ ( ) [ ]]
e (w + αt)T−1(z + α)(aτ + b)a · e rtTrτ + (w + αt)r + rt(z + α) (aτ + b) + 2Re σtr
[ (4.68) ]
·e 2Re[σt(aT−1α)− bstα] .
(4.69)
From this our first transformation law follows.
Now we consider the second of these transformation laws. To prove this we make
another change of variables in our sum ϕ(w, z)ψ(w, z). Note that
( √ ) √ √ ( √ ) √
d(T )T−1 = ± det i DT (i DT )−1 or ± i D det i DT (i DT )−1 (4.70)
depending on whether n is odd or even. In any event we see that En(D)d(T )T−1β ∈
On. Thus cT−1β ∈ On. We exchange r for r − cT−1β.
35
After this change of variables we proceed by direct comparison of the sums in
Expressions 4.60 and 4.61. Since JMτ = −cτ−d , it suffices to show
aτ+b
t t
(β r + a(β T−1z + wT−1
t
β + β T−1β(JMτ)) + rtβ))(−cτ − d)
(4.71)
≡ − t t t t( c(β T−1β) + rtβ + β r + awT−1β + aβ T−1z)(−cτ) + b(wT−1β(−c) + (−c)(β T−1z))
(4.72)
− t t − t td(r β + β r) β T−1z − wT−1β − β T−1βJMτ (mod Z)
(4.73)
The difference of these two modulo Z is tdcβ T−1β which, by Lemma 4.5, is an integer.
Next our goal is to relate these functions ψ and ϕ to our theta functions. The
basic idea is to study the Fourier expansion of the product and compute the Fourier
coefficients. From here we can use a change of variables to relate our function ψ to
θ.
L[emma 4.10. Let s
′ = bTs− ]aσ and σ
′ = dTs− cσ. Then we have
e s′
t
T−1s∑′JMτ +[ 2Re[s′T−1σ′] ϕ(w, z) · ψs,σ(w, z) (4.74) ]t t t
= γ(0) e (s′ + g) T−1(s′ + g)JMτ + wT−1(s′ + g) + (g + s′) T−1z + 2Re[(s′ + g) T−1σ′
n
g∈(O#)
(4.75)
for some constant γ(0).
Proof. Let
F (w, z) := ϕ(w, z)ψs,σ(w, z). (4.76)
We follow the following steps:
36
1. Define a new function F̃ and check the transformation
F̃ (w + αt, z + α) = F̃ (w, z)
for all α ∈ TOn.
2. Write out the Fourier expansion of F̃ and translate this to a series expansion
of F .
3. Evaluate tF (w+ β JMτ, z+ βJMτ) in two ways. First evaluate directly, using
step (2).
4. Use the equality of 4.60 and 4.61 tF (w + β JMτ, z + βJMτ) and perform the
change of variables g 7→ g + β on the sum.
5. Compare the series in (3) and (4) and plug in g = 0 to get a formula for the
coefficients.
6. Use (5) and the series expansion of F to prove the equality of Expressions
4.90 and 4.91.
Step 1: Let F (w, z) = ϕ(w, z)ψ(w, z). Define
[ ]
t
F̃ (w, z) = e −(wTs′ + s′ Tz) F (w, z). (4.77)
The transformation law for F̃ follows from Equation (4.59).
Step 2: We know F̃ is holomorphic on Cn × Cn because both ϕ and ψ are. By Step
1 we know that F̃ is periodic with respect to the lattice
{ }
(αt, α) : α ∈ TOn . (4.78)
37
Because F̃ is holomorphic F̃ will have a Fourier expansion over the dual lattice of
the form
∑
F̃ (w, z) = γ(Tg)e[wg + gtz]. (4.79)
g∈T−1(O#)n
Then, after the change of variables g 7→ T−1g, we have
∑ [ ]
−1 ′ ′ tF (w, z) = γ(g)e wT (s + g) + (s + g) T−1z . (4.80)
( g)∈(O#)n
Step 3: Let β ∈ O# n. Plugging in t(w + β JMτ, z + βJMτ) gives
∑ [ ] [
t
F (w + β JMτ, z + βJMτ]) (4.81)
t t
= γ(g)e 2Re[(s′ + g) T−1β]JMτ e wT−1(s′ + g) + (s′ + g) T−1z . (4.82)
n
g∈(O#)
Step 4: Note
dstβ − cσtT−1 tβ = σ′ T−1β. (4.83)
By Equation (4.59) we have, [ ( )]
t t t t
F (w + β JMτ, z + βJMτ) = e − β T−1βJMτ + wT−1β + β T−1z + 2Re[σ′ T−1β] F (w, z).
(4.84)
Using the series expansion of F and replacing the index g by g + β gives
t
F (w + β JMτ, z + βJMτ)
∑ [ ( )] [ (4.85) ]t t t
= γ(g + β)e − β T−1βJMτ + 2Re[σ′ T−1β] e wT−1(s′ + g) + (s′ + g) T−1z .
g∈(O#)n
(4.86)
38
Step 5: Comparin∑g 4.82 and[4.86 [we see ] ]t − [ − ]γ(g)e 2Re (s′ + g) T 1β JMτ e wT 1g + gtT−1z (4.87)
∑ ng∈(O#) [ ]
− t
[ ]
= γ(g + β)e β T−1βJMτ − 2Re t[σ′ T−1β] e wT−1g + gtT−1z . (4.88)
n
g∈(O#)
If we compare the g =[(0 coefficients then we g)et ]
t
γ(β) = γ(0)e β T−1 Re ′tβ + 2 [s T−1β] JMτ + 2Re ′t[σ T−1β] . (4.89)
[ Step 6: If we combine Equ]ation (4.89) with 4.80 we get
t
e s′ T−1s∑′JMτ +[ 2Re[s′T−1σ′] ϕ(w, z) · ψs,σ(w, z) (4.90) ]t
= γ(0) e (s′ + g) T−1(s′ + g)JMτ + wT−1(s′
t
+ g) + (g + s′) T−1z + 2Re[(s′
t
+ g) T−1σ′
n
g∈(O#)
(4.91)
where
σ′ = dTs− cσ and s′ = bTs− aσ. (4.92)
as desired.
Let [ ( )]
t
η = η(τ, s, σ) = e − s′ T−1s′JMτ + 2Re t[s′ T−1σ′] γ(0). (4.93)
Lemma 4.10 then becomes
ϕ(w, z)∑· ψs,σ(w[, z) (4.94) ]
′ t= η e (s + g) T−1(s′
t t
+ g)JMτ + wT−1(s′ + g) + (g + s′) T−1z + 2Re[(s′ + g) T−1σ′
n
g∈(O#)
(4.95)
In the next lemma we compute the value of η. In the process we’ll run into the
following Gauss sum.
39
Definition 4.11. For T and a ∈ Z non-z∑ero , de[fine ]stTs
G(T, a) = e .
a
s∈On/aOn
We’ll also need the following lemma from Haverkamp’s thesis [Hav95]:
Lemma 4.12. For ∫c ∈ R and r, q ∈ C with Im(q) < 0 we have[ ] [ ]
e −qz2 + rz dz = (2iq)−1/2e r2/4q (4.96)
Im(z)=c
Lemma 4.13. We have
( 1 ) 1 [ ]η = √ G(bT, a)e −bdstn Ts− acσtT−1σ + 2adRe[σtTs] .
det −i DT (a(aτ + b))
(4.97)
Proof. We start by mu[ltip(lying both sides of equation from)4].10 by
− ′t −1 −1 ′ ′te s T sJMτ + wT s + s T−1z .
We write the left hand side as ∑
e[R(q, w, z)]
q∈On
where ( )
R(q, w, z) = wT−1
t t
z(aτ + b)(a) + (s+ q) T (s+ q)τ + w(s+ q) + (s+ q) z (aτ + b)
(4.98)
t t
+2Re[σt(s+ q)]− s′ T−1s′JMτ − wT−1s′ − s′ T−1z.
(4.99)
The righ∑t hand side[(will be ) ]
t
η e ptT−1p− s′ T−1s′ JMτ + (wT−1(p− s′) + (p− s′)tT−1z + 2Re[ptT−1σ′] .
n
p≡s′ (mod (O#) )
There are two transformation laws that R satisfies: Let g, h ∈ Cn
40
1.
( )
t
R(ag+h,w, z) = R(h,w, z)+ a2gtTg + 2Re[(ag) T (s+ h)] τa(wg+gtz)(aτ+b)+2aRe[σtg].
2.
R(h,w+gtT, z+Tg) = R(h,w, z)+2aRe[σtg]+(gtz+wg+gtTg)(a)(aτ+b)+2aRe[gtT (s+b)]τ.
Both of these follow from direct computation.
From (1) and (2) above together we see that, for g, h ∈ On
R(ag + h,w, z) ≡ R(h,w + gtT, z + Tg) (mod Z).
In order to compute η we set z = wt and integrate both the left and right hand
sides over Cn/ (OnT ). We first consider the right side.
The right side is invariant under translation of w by α ∈ OnT . Our integral is
∑
η I(p) (4.100)
p≡s′ (mod (O#)n)
where ∫ [ ]
I(p, s′) := e (wT−1(p− s′) + (p− s′)tT−1wt + 2Re[ptT−1σ′] dw. (4.101)
Cn/OnT
Note that if we substitute w + z for w we get
I(p, s′
[ ]
) = e 2Re[(p− s′)T−1zt] I(p, s′) (4.102)
for any z ∈ Cn. Hence I(p, s′) = 0 unless p = s′. After plugging in p = s′ our right
hand side becomes.
[ ]
ηe 2Re t[s′ T−1σ′] vol (OnT ) . (4.103)
41
We have (√ )n
D
vol (OnT ) = det(T )2vol(On) = det(T )2 . (4.104)
2
Thus our right hand side is (√ )n
D [ ]t
η det(T )2 e 2Re[s′ T−1σ′ (4.105)
2
∫Next we∑compute the left han∫d side. W∑e have[ ]
e[R(t, w, wt)] = ∫ e R(h,w + g
tT,wt + Tg) (4.106)
Cn/OnT p∈On C/∑OnT h∈On/aO
= e[R(h,w,wt)] (4.107)
h∈O∑n/aOn ∫C [ ( )]t(h+ s) T T (h+ s)
= e R h,w − , wt − dw
a a
h∈On/aOn C
(4.108)
Recall that a ̸= 0. Now we need to compute this integral. After expanding R and
simplifying the resul∫ting [expression breaks into two pieces: ]
e wT−1wt(a)(aτ + b)− 2Re[wT−1s′] dw (4.109)
C
∑ [ ]t
Re t − ′t −1 2 (s+ h) T (s+ h)e 2 [σ (s+ h)] s T s′JMτ + Re[s′(h+ s)]− b .
a a
h∈On/aOn
(4.110)
First we’ll simplify (4.110)[. Our sum in (4.110) is]
b
e st
t
Ts− s′ T−1s′JMτ G(bT, a). (4.111)
a
Now we consider (4.109). Since T is Hermitian we can find another Hermitian matrix
G such that G2 = T−1. Let v = wG and u = Gs′. Our integral is, after making the
substitution w 7→ wG−1 ∫ [ ]
det(T ) e vvt(a)(aτ + b)− 2Re[vu] dv. (4.112)
Cn
42
If we break this integral into components and write v and u for vi and ui respectively
we’ll find [∫ [ ] ]n
e (v)2(a)(aτ + b)− 2Re[vu] dv . (4.113)
C
Let v = x∫+ iy and u = p+ iq so that o∫ur integral split into[ ] [ ]
e x2(a)(aτ + b)− 2xp dx · e y2(a)(aτ + b)− 2yq dy. (4.114)
R R
We view R = {z ∈ C : Im(z) = 0} so we can apply Lemma 2.6. (b) from
Haverkamp’s thesis, Lemma 4.12 in this paper, to each of the above. We get that
the above integral is equal to [ ]
i p2− + q
2
e . (4.115)
2(a)(aτ + b) a(aτ + b)
In total we find that our(integral in (4).110)[is equal ton ]
i 1 t
det(T ) e − s′ T−1s′ . (4.116)
2(a)(aτ + b) (a)(aτ + b)
Combining 4.109 and 4.110 give the expression for the le(ft hand side as[ ] )ni
e bdstTs− 2cbRe[stσ] + acσtT−1σ G(bT, a) det(T ) . (4.117)
2a(aτ + b)
Setting expres(sions (4.116) and)(4.105) equal and solving for η givesn
1 [ ]
η = √ i G(bT, a)e −bdsTs− acσtT−1σ + 2adRe[σtTs] .
det(T ) D(a)(aτ + b)
(4.118)
A slight simplification of the above gives the desired result.
Using this we can get a transformation for our theta function which, after
analyzing the sum G will be that given in Proposition 4.8.
43
Proposition 4.14. We have, for M ∈ Γ0(En(D)d(T )),
θT,s|T,n(MJ) = ∑ [ ( )] (4.119)( 1 ) · 1√ G(bT, d) e a 2Re[stT−1s′] + bstT−1s · θn T,s′ .
det −i DT d ′ ns ∈(O#) /TOn
(4.120)
Proof. (Plugging in our )value for η to Equation (4.10) gives
wT Tz
ψT−1s,0 , = ∑ (4.121)aτ + b aτ + b1 1 [ ( )]√ · G(bT, a) e d 2Re[stT−1s′]− bstT−1s · θm,s′ |T,n[−JM ].
det(−i DT ) an n
s∈(O#) /TOn
(4.122)
Recall we have ∑ [ ]
ψs,σ(w, z) := e r
tTrτ + (wr + rtz)(aτ + b) + 2Re[σtr] . (4.123)
r≡s (mod On)
Using this and the change of (variables r 7→ T−)1r gives
wT Tz
ψT−1s,0 , = θT,s(τ, w, z). (4.124)
aτ + b aτ + b
After slashing both sides by M−1J and replacing M−1 by M we get, for any M ∈
Γ0(En(D)d(T ))
θT,s|T,n(MJ) = ∑ [ ( )] (4.125)( 1 1√ ) · G(−bT, d) e a 2Re[stT−1s′] + bstT−1s · θn T,s′ .
det −i DT d
s′
n
∈(O#) /TOn
(4.126)
Our goal over the next few results is to show that G(bT,d)n is a character modulod
En(D)d(T ) and that it does not depend on b. First we show that it is a group homomorphism
44
from the group Γ0(En(D)d(T )) to C×. In the process we’ll need to compute some
exponential sums that appear, for which we need the following lemmas.
Lemma 4.15. Let A ∈ Mn(O). Then [On : AOn] = Nm(det(A)).
Proof. See [Cla14, Lemma 1.15] for a proof of this claim. Here R = O, M = T and
Λ = O.
Lemma 4.16. We have
∑ [ [ ]] 0 if s′ ̸= 0
e 2Re stT−1s′ =  √ (4.127)
n 
s∈(O#) /TOn Nm(det(i DT )) if s′ = 0.
Proof. Suppose that s′ ≠ 0. I claim 2Re(s T−1s′0 ) ̸∈ Z for some s0. Since we
know s′ ̸∈ TOn we must have T−1(s′) ̸∈ On. We can find s ∈ (O#)n0 such that
2Re[s tT−1s′0 ] ̸∈ Zn because O is the dual lattice of O#. With this choice of s0 in
mind we have [ ] ∑ [ ]
e 2Re(s −1 ′0T s ) e 2Re(sT−1s′) (4.128)
∑ [ ns∈(O#) /TOn ]
= e 2Re(s′T−1(s+ s0) (4.129)
∑ns∈(O#) /TOn [ − ′ ]= e 2Re(sT 1s ) . (4.130)
n
s∈(O#) /TOn
Since e [2Re(s T−10 s′)] ≠ 1 we must have that the sum is zero. If s′ = 0 then the
result follows from Lemma 4.15.
a b
Lemma 4.17. For M =   ∈ Γ0(En(D)d(T )) let
c d
G(bT, d)
χ (M) = . (4.131)
dn
Then χ(MM ′) = χ(M)χ(M ′).
45
Proof. This result follows by applying Proposition 4.14 to θT,0. By Proposition 4.14
we have
1 ∑
θT,0|T,nMJ = √ χ(M) θ− T,s
′ . (4.132)
det( i DT )
s′
n
∈(O#) /TOn
Thus
∑
θT,0|
1
T,nM = √ χ(M) θ− T,s
′|T,n(−J). (4.133)
det( i DT )
s′
Now again by Proposition 4.14 we have
1 ∑ [ ]
θT,s′|T,n(−J) = √ (−1)nG(−T,−1) e −2Re[s′′T−1s′] θ− T,s
′′ . (4.134)
det( i DT )
s′′
Direct computation from the definition gives G(−T,−1) = 1. In total we have
(−1)n ∑∑ [ ]
θT,0|M = √ χ(M) e −2Re
t
[s′ T−1s′′] θT,s′′ . (4.135)
det(−i DT )2
s′ s′′
By Lemma 4.16 we have
| (−1)
n √
θT,0 M = √ χ(M)Nm(det(i DT ))θT,0 (4.136)
det(−i DT )2
= χ(M)θT,0. (4.137)
Since θT,0 ≠ 0 and the slash operator is multiplicative the desired result follows.
Next we want to compute χ(M) in the case that d is coprime to 2Dd(T ). The
idea is to try and diagonalize bT modulo d and then reduce to the one dimensional
case, which is solved by Haverkamp.
Lemma 4.18. We have for d coprime to 2Dd(T ) and b coprime to d,
G(bT, d)/dn = (χD(d))
n (4.138)
46
Proof. Recall
∑ [ ]stTs
G(T, a) = e . (4.139)
a
s∈On/aOn
We first reduce to the case when d = pr. We show that for coprime integers p and q
we have
G(bT, pq) = G(pT, q)G(qT, p). (4.140)
Note that (x, y) 7→ px + qy gives a bijection from On/pOn × On/qOn to On/pqOn.
Thus
∑ [ ]stTs
[ G(T, pq) = e (4.141)∑ ∑ pqs∈On/pqOn ]xt(qT )x yt(pT )y
= e + + 2Re[xtTy] . (4.142)
p q
x∈On/p y∈On/q
The result then follows from the fact that, since T ∈ Λn(O) and x, y ∈ On we have
2Re[xtTy] ∈ Z (4.143)
by Lemma 3.2. We can then reduce to the case when d = pr by splitting d into
prime factors in this way.
√
Since d is coprime to D we see that i D is an invertible element of O/dO. If T
√
does not have entries in O then we can replace T by DT by replacing s with i Ds
in our sum for G. We assume that T has entries in O.
Next I claim that we can find P ∈ GL tn(O/dO) such that P TP is diagonal modulo
d. We prove the result by induction on the dimension n. The n = 1 case is clear.
Now suppose that we know that for any n × n half-integral Hermitian matrix R ∈
Mn(O) with determinant coprime to p we can find an invertible G such that
t
G RG
is diagonal modulo d. Using our induction hypothesis it suffices to show that T is
47
equivalent to a matrix  ℓ 0 (4.144)
0 R
where R is an n× n Hermitian matrix.
First we argue that we can assume one of the diagonal entries of T is coprime to
p. To do this we construct a vector v such that vtTv is coprime to p. We’ll then
extend {v} to a basis for On/dOn to get a matrix tP such that P TP has the first
diagonal entry coprime to p. Suppose that all the diagonal entries are divisible by
p. I claim that since tdet(T ) is coprime to p we can find P such that P TP has an
off diagonal entry aij such that |a 2ij| is coprime to p.
If p is inert we can take P = In+1. This immediately follows from the fact that
det(T ) is coprime to p. Suppose that p = pp. Let Q = {σ ∈ S −1n : σ = σ }. Then,
after pairing the terms corresponding to σ and σ−1 we have
∑ ∏ ∑ ∏ ∑ [∏ ]n n n
det(T ) = (−1)σ t σiσ(i) = (−1) t σiσ(i) + (−1) 2Re tiσ(i) .
σ∈Sn i=1 σ∈Q i=1 σ ̸∈Q i=1
∏ (4.[1∏45) ]
Since det(T ) is coprime to p we must have either n −1 ni=1 tiσ(i) for σ = σ or 2Re i=1 tiσ(i)
for σ ̸= σ−1 not lie∏in p.
First suppose that n −1i=1 tiσ(i) is coprime to p with σ = σ . Since tii is divisible
by p by assumption we can find i with σ(i) ≠ i. Then |t 2iσ(i)| divides the above
product and hence mus[t∏be coprim] e to p.
Now suppose that 2Re ni=1 tiσ(i) is coprime to p for some σ ̸= σ−1. Note that
none of the tiσ(i) can lie in pO. If for some i tiσ(i) ̸∈ p, p then we’ll have |t 2iσ(i)|
coprime to p and we’re done. Consider when tiσ(i) ∈ p or p for each i. Since σ ̸=
σ−1 we can choose i0 with σ(i0) ̸= σ−1(i0). Without loss of generality assume
48
ti σ(i ) ∈ p. Then ti σ−10 0 (i ) ∈ p. Now choose P to be the elementary matrix which0 0
replaces column σ−1 t(i0) by column σ−1(i0) added to column σ(i0). Then P TP will
have (i0, σ−1(i0)) entry ti ,σ(i ) + ti ,σ−10 0 (i ). Since ti,σ(i) ∉ pO = p ∩ p we see that0 0
ti0,σ(i ) + ti ,σ−10 (i ) does not lie in p or p and hence has norm coprime to p. Thus in0 0
any event we can conjugate by an invertible matrix P and assume that T has an
entry tij with |t 2ij| coprime to p.
Let v = [vi] be a vector with vi = tij and vj = 1 and all other entries zero.
Then
vtTv = |t |2t + 2|t |2ij ii ij + tjj (4.146)
which is coprime to p. The set {v} ∪ {ek}k ̸=j will give a basis for On/prOn. Let P
be the matrix with columns v, b2, . . . bn where b2, . . . , bn are the basis vectors chosen
above. Then the first entry of tP TP is coprime to p, as it is equal to vtTv.
Thus in any event we can assume one of the diagonal entries of T is coprime to p.
If one of the other diagonal entries is coprime, say tii to p we can move it to the
front by reordering our basis. Thus we can assume t11 is coprime to p. Using this
and conjugation by transvections (ti,j(a) = In+1 + aEij with i ̸= j) we can eliminate
the top row, and since our conjugation preserves the Hermitian property, it will
also eliminate the first column as well. To do this note that after conjugation by
t1,j(a) our matrix has entry (1, j) entry given by
t1,j + at1,1. (4.147)
Since t1,1 is coprime to p we can choose a so that this is congruent to zero. Since
this conjugation does not interfere with entries left of (1, j) we can progressively
reduce the entire first row to zero (except the very first entry). Then we can apply
the induction hypothesis to get the desired result.
49
After a change of variables by our matrix P we can assume that T is a diagonal
matrix. Note that each of the diagonal entries must be coprime to p. Let mi be the
ith diagonal entry of T . We can break our sum defining G into a product of n sums
of the form ∑ [ ]miN(si)
e . (4.148)
d
si∈O/dO
Now as long as d is odd, by [∑Hav95,[Lemma .]04], we havemiN(si)
e = χD(d)d. (4.149)
d
si∈O/dO
From this the result follows.
Now we prove Proposition 4.8.  
a b
Proof of Proposition 4.8. Let M =   ∈ Γ0(En(D)d(T )). By Proposition 4.14
c d
it suffices to show χ(M) = χD(d)n. This follows immediately from Lemma 4.18
when (d, 2D) = 1.Suppose that d is not coprime to 2D. Let s be the product of all
prime factors of 2D not dividing d. Note that when the lower right entry of M ′ is 1
χ(M ′) = 1. Then    1 + sd(T ) s ∗ ∗ 

χ(M) = χ(M)χ = χ  . (4.150)
d(T ) 1 ∗ sc+ d
Let p divide 2D. If p divides d then it doesn’t divide c because ad − bc = 1 and
it doesn’t divide s by construction. Hence in this case p doesn’t divide sc + d.
If p doesn’t divide d then it must divide s by construction and hence sc. Since p
doesn’t divide d it doesn’t divide sc + d. Thus sc + d is coprime to 2D. We see
χ(M) = χD(sc + d)
n. If n is even then this is 1 because χD is a real character so
χ(M) = 1χD(d)
n. If n is odd then D divides c and so χD(sc + d)n = χ nD(d) .
Proposition 4.8 then follows.
50
4.3 Linear Independence of Theta Functions
One of the important properties we’d like to know about our theta functions
is that they’re linearly independent. This allows us to conclude that if we have two
theta expansions that are equal, they must have identical theta coefficients, which
will become useful in the next chapter.
Proposition 4.19. We∫have, for s, s′ ∈ (O#)n with s ̸≡ s′ (mod TOn),
θT,s(τ, w, z)θT,s′(τ, w, z)dwdz = 0. (4.151)
Pnτ
Here
Pτ := {(α + βω + γτ + δωτ, α + βω + γτ + δωτ) : 0 ≤ α, β, γ, δ < 1} ⊂ C× C
(4.152)
and O = Z+ ωZ.
Proof. Let ∑ [ ]
θT,s(τ, w, z) = e σ
tT−1στ + wσ + σtz (4.153)
σ∈(O#)n
σ≡s (∑mod TOn) [ − ]θ t 1 tT,s′(τ, w, z) = e ρ T ρτ + wρ+ ρ z . (4.154)
ρ∈(O#)n
ρ≡s′ (mod TOn)
After exchanging the order o∑f sum[mation and integra]tion integral 4.151 we get
e σtT−1σ − ρtT−1ρτ I (4.155)
σ≡s
ρ≡s′
where ∫ [ ]
I = e wσ + σtz − ρtwt − ztρ dzdw. (4.156)
Pnτ
Consider the substitution w 7→ w + ωzt, z 7→ wt + ωz. If we call this substitution
H : Cn × Cn → Cn × Cn then the equations below follow from direct computation:
51
• det(H) = Dn/2
• H−
( )
1 n(P nτ ) = (C/Z+ τZ)
2 .
Then ∫
I = Dn e (w(2Re[σ]) + 2Re[ωσ]z − w(2Re[ρ])− 2Re[ωρ]z) dwdz.
n
((C/Z+τZ)2)
(4.157)
If we wr∫ite w = x1 + iy1 and z = x2 + iy2 then we get
I = Dn e [x1 (2Re[σ]− 2Re[ρ])] e [(2Re[ωσ]− 2Re[ωρ])x2] dx1dx2e [f(y1, y2)] dy1dy2
(4.158)
for some unimportant linear function f . We see, after some potential shifting of the
dom∫ain, the integrals with respect to ∫x1 and x2 are
e [x1 (2Re[σ]− 2Re[ρ])] dx1 · e [(2Re[ωσ]− 2Re[ωρ])x2] dx2. (4.159)
[0,1]n [0,1]n
Since s ̸≡ s′ (mod TOn) we must have σ ̸≡ ρ (mod TOn) which means in particular
that σ ̸= ρ. We must have either 2Re[σ] ̸= 2Re[ρ] or 2Re[ωσ] ̸= 2Re[ωρ] since
otherwise σ = ρ. Since σ, ρ ∈ (O#)n we must have 2Re[σ], 2Re[ωσ], 2Re[ρ], and 2Re[ωρ]
must all lie in Zn. Thus I = 0 as desired and the result follows.
Corollary 4.20. The collection {θs}s∈(O#)n/TOn is linearly independent over C.
Proof. Suppose we have
∑
csθs = 0. (4.160)
( ) s∈(O#)n/TOn
Then for any s′ ∈ O# n /TOn∫we have∑ ∫
0 = cs θsθs′dwdz = c
2
s′ |θs′ | dwdz. (4.161)
s Pτ Pτ
52
Note that θs ≠ 0 on Pτ , plug in (i∫, 0, 0) for example, so that
|θs′|2dwdz ≠ 0 (4.162)
Pτ
and so we must have cs′ = 0 as desired.
53
CHAPTER 5
THE EICHLER--ZAGIER MAP FOR HERMITIAN JACOBI FORMS
5.1 Eichler--Zagier Map
Now that we have theta expansions we can define an Eichler--Zagier map. This
will take our Hermitian-Jacobi form and give us a classical modular form. This will
be a generalization of a similar map from [EZ85]. We give a brief overview of the
original Eichler--Zagier map.
Given a classical Jacobi form of index m ϕ Eichler and Zagier give a series expansion
of the form
∑
ϕ(τ, z) = hµ(τ)θm,µ(τ, z). (5.1)
µ (mod 2m)
Each of the functions hµ is a classical modular form of half integer weight and the
map
∑
ϕ 7→ hµ(4mτ), (5.2)
µ (mod ()2m)
the original Eichler--Zagier map, gives an isomorphism from the space of Jacobi
forms of index m and weight k and a particular subspace of modular forms of weight
k−1 . We generalize this map here and explore some of the relationship between our
2
Hermitian Jacobi forms and classical modular forms
First we need to translate the transformation law of our theta function into a transformation
law for our coefficients. Throughout let ϕ be an Hermitian Jacobi form of degree n,
weight k and non-singular index T with theta expansion
∑
ϕ = θT,shs. (5.3)
s∈(O#)n/TOn
54
Let Θ = [θs]s∈(O#)n/TOn and H = [hs](O#)n/TOn so that ϕ = Θ
tH. We can define
slash operators component-wise on these vector-valued functions. Using the definitions
directly gives
( )
ΘtH |T,kM = (Θ| tT,nM) H|k−nM (5.4)
for any M ∈ SL2(Z). If we translate Proposition 4.8 into a law for Θ we find that if ( 1 ) [ ( − [ ])]

UT (MJ) = √ χD(d)ne a bstT 1s+ 2Re stT−1s′  (5.5)
det −i DT
s,s′
then
Θ|T,nMJ = UT (MJ)Θ (5.6)
for any M ∈ Γ0(En(D)d(T )).
We also have ∣∣∣  ∣∣ 1 1θ  [ ]= e st −1s T s θs (5.7)
T,n 0 1
so that
 1 1 [ [ − ′]]UT = δ e st 1s,s′ T s s,s′∈ . (5.8)(O#)n/TOn
0 1
These two computations show that for every M ∈ SL2(Z) there exists some
matrix UT (M) such that Θ|T,nM = UT (M)Θ.
Our first goal is to show UT (M) is always unitary.
Lemma 5.1. Let M ∈ SL2(Z). Then UT (M) is unitary.  
1 1
Proof. Since UT is multiplicative, it suffices to check that UT (J) and
 UT 
0 1
are unitary, since these matrices generate SL2(Z).
55

1 1
The fact that  

UT  is unitary follows from the fact that T is Hermitian.
0 1
t
Consider UT (J). The i, j entry of UT (J) UT (J) is
( 1 ) ∑ [ ]√ e 2Re((si − s −1j)T sk) . (5.9)
| det DT |2 n
1≤k≤|(O#) /TOn|
By Lemma 4.15 the diagonal entries are all 1. To show that the off diagonal entries
are 0 apply Lemma 4.16. Thus UT (J) is unitary as desired.
Proposition 5.2. We have, for M ∈ Γ (E (D)d(T )),
(χ (d))n ∑
0 n
[
h | DMJ = √ e −a(bstT−1s+ 2Re[stT−
]
1 ′
s k−n s ]) hs′ . (5.10)
det(i DT ) n
s′∈(O#) /TOn
Proof. Since ϕ|T,kMJ = ϕ we have
ΘtH = (Θ| MJ)tT,n (H| tk−nMJ) = Θ (UT (MJ))t(H|k−nMJ). (5.11)
By the linear independence of theta functions, Corollary 4.20, we see that
(UT (MJ))
−tH = H|k−nMJ. (5.12)
By Lemma 5.1 we know UT (M) is unitary for any M ∈ SL2(Z). Thus Equation
(5.12) gives
H|k−nMJ = UT (MJ)H (5.13)
and translating this to each hs gives the desired result.
Lemma 5.3. If x ≡ y (mod En(D)d(T )) then we have
stT−1sx ≡ stT−1sy (mod Z). (5.14)
Proof. This follows immediately from Lemma 4.5
56
Corollary 5.4. Let ϕ be an Hermitian Jacobi form of degree n, invertible index T
and weight k. Let hs be a theta coefficient of ϕ. Then hs ∈ Sk−n(Γ(En(D)d(T ))).
Proof. First we address holomorphy concerns. We know that hs is given by a Fourier
series that converges absolutely and uniformly on compact subsets of H. This gives
us holomorphy on H and boundedness of hs at infinity. To get holomorphy at other
cusps recall that, for H = [hs] our vector of theta coefficients we have, for any
t
M ∈ SL2(Z) H|k−nM = UT (M) H. We then see that hs|k−nM has a similar Fourier
expansion (given as some linear combination of the other theta coefficients) and is
thus bounded at infinity as well. Since T is non-singular the Fourier expansions of
each hs has no constant term. Hence each hs is a cusp form.  
a b
The important part is the transformation law. For M =   ∈ Γ(En(D)d(T ))
c d
we have, by Proposition 5.2,
n ∑ [ ]
hs|
χD√(d)k−nMJ = e −a(bstT−1s+ 2Re[stT−1s′]) hs′ . (5.15)
det(i DT )
s′
If n is even then χ (d)nD = 1 since χD(d) = ±1. If n is odd then d ≡ 1 (mod D)
and χD is a chara[cter(modulo D we must hav)e]χD(d[) = 1. By Lem]ma 5.3 we have
e −a bstT−1s+ 2Re[stT−1s′] = e −2Re[stT−1s′] . (5.16)
Using Proposition 5.2 we see ∑ [ ]
hs|
1
J = √ e −2Re[stT−1s′] h′s. (5.17)
det(i DT )
s′
Hence hs|MJ = hs|J so hs|M = hs.
Theorem 5.5. Let ϕ be an Hermitian modular form of weight k, degree n ≥ 1 and
non-singular index T . Then ϕ has a the∑ta expansion of the form
ϕ(τ, w, z) = hs(τ)θs(τ, w, z) (5.18)
n
s∈(O#) /TOn
57
where each hs is a classical modular form.
Proof. The existence of this expansion is Proposition 4.1 and the fact that each hs
is a classical modular form is Proposition 5.4.
Using this we can prove the following corollary.
Corollary 5.6. The space of Hermitian Jacobi forms of degree n and invertible
index T is finite dimensional.
Proof. We have an injective map of C vector spaces
⊕
H : J nT,k → Sk−n(Γ(En(D)d(T ))) (5.19)
s∈(O#)n/TOn
defined by H(ϕ) = [hs]s∈(O#)n/TOn .⌊We know ∏ ⌋
(k − n)(En(D)d(T ))3 p|E (D)d(T )(1− 1/p2)
dimC(Mk−n(Γ(En(D)d(T ))) =
n
12
(5.20)
so we get
( ) ∣ ∣ ⌊ ∏ ⌋(k − n)(En(D)d(T ))3 (1− 1/p2)
dim J nT,k ≤ ∣ O# n p|E (D)d(T )( ) /TOn∣ n⌊ 1∏2 ⌋ (5.21)
(k − n)(En(D)d(T ))3 p|E (D)d(T )(1− 1/p2)
= Dn det(T )2 n (5.22)
12
We define our Eichler--Zagier map and prove it is well-defined.
Definition 5.7. Define ι : J n nT,k → Sk−n(Γ0(En(D)d(T )), χD) by∑
ι(ϕ)(τ) = h(τ) := hs(En(D)d(T )τ). (5.23)
n
s∈(O#) /TOn
We argue that ι is well-defined.
58
Proposition 5.8. If we define ι as above then ι(ϕ) ∈ S nk−n(Γ0(En(D)d(T )), χD) for
ϕ ∈ J nT,k.
Proof. Let ϕ be an Hermitian Jacobi form of degree n and invertible index T . Let
a b
hs be a theta coefficient of ϕ and
  ∈ Γ0(En(D)d(T )). We have
   En(D)d(T )c d a b ∑ a En(D)d(T )b
h| k−n   = hs|k−n   (En(D)d(T )τ). (5.24)
En(D)d(T )c d s c d
In order to use the above result note that   a En(D)d(T )b  −d c = J  J. (5.25)
c d bEn(D)d(T ) −a
Using this and the transformation for each hs gives
h|k−nM
χ (d)n ∑∑∑ [ ] (5.26)D
= √ e −2Re[stT−1s′ t t] + dcs′ T−1s′ + 2dRe[s′ T−1s′′] hs′′ .
det( DT )2 s s′ s′′
(5.27)
We see that, computing the sum over s, this inner sum will be zero unless s′ ≡ 0, in
√
which case it is det( DT )2. Simplifying gives the desired transformation for h.
The following results give an important subspace of J nk,T on which the Eichler-
-Zagier map is injective, and introduce a family of maps between different spaces of
Hermitian Jacobi forms. These maps give us some additional relationships between
spaces of Hermitian Jacobi forms and generalize useful constructions in [AD19].
Definition 5.9. Let k ∈ Z≥0, n ∈ Z>0 and T ∈ Λ+n (O). Define a subspace
J n,spez nk,T ⊂Jk,T (5.28)
59
consisting ofthose ϕ ∈ J nk,T whose Fourier coefficients α(m,r)dependonly onm r 
det . That is ϕ ∈J n,spezk,T if, whenever dm r m′ r′= d ,t
rt T rt T r′ T
we also have α(m, r) = α(m′, r′).
Proposition 5.10. The Eichler--Zagier map is injective on J n,spezk,T .
Proof. Let
∑
ϕ = θ n,spezT,shs ∈Jk,T (5.29)
( )s∈(O#)n/TOn
be non-zero. Choose s ∈ O# n so that hs ̸= 0.[Recall the Fo]urier expansion of hs∑ N
hs(τ) = αs+TOn(N)e τ . (5.30)
En(D)d(T )
N∈IT,s
Choose N such that hs has non-zero Nth Fourier coefficient.
Since ϕ ∈ J n,spezk,T we have α # ns+TOn(N) = αr+TOn(N) for any r ∈ (O ) and N ∈
IT,s. Denote this common value by α(N). Then we have
∑ ∑
ι(ϕ) = α(N)e [τ ] . (5.31)
r∈(O#)n/TOn N∈IT,r
From this we see that the Nth Fourier coefficient of ι(ϕ) is
∣{ ( ) }∣
α(N) · ∣ r ∈ O# n : N ∈ I ∣T,r . (5.32)
Since α(N) ≠ 0 and s ∈ {r : N ∈ IT,r} we see that ι(ϕ) has non-zero Nth Fourier
coefficient and is thus non-zero.
In [AD19, Proposition 4.8] Anamby and Das prove that this is the maximal
subspace of J nk,T on which the Eichler Zagier map is injective when n = 1 under
some additional conditions on T and our quadratic imaginary field K.
60
Definition 5.11. Let P ∈ Mn(O) be a matrix with non-zero determinant. Define,
for ϕ ∈J nk,T a function ϕ|UP ∈J n t defined byk,PTP
t
ϕ|UP (τ, w, z) := ϕ(τ, wP, P z). (5.33)
One can check that tPTP ∈ Λn(O) and that ϕ|UP has the correct transformations.
The Fourier expansion is discussed in the following proposition.
Proposition 5.12. Let ϕ ∈J nk,T have Fourier expansion∑
ϕ(τ, w, z) = α(m, r)e(mτ + wr + rtz). (5.34)
 m∈Z,r∈(O#)n


m r 
 ∈Λ+n+1(O)
rt T
Then we have
∑
ϕ|UP (τ, w, z) = β(m, r)e(mτ + wr + rtz). (5.35)

m∈Z,r∈(O#)n
m r 

∈Λ+n+1(O)
rt T
where
0 if r ∉ P (O#)n
β(m, r) :=  ( ) (5.36)α(m, r′) if r = Pr′ for nr′ ∈ O# .
Note that β is well-defined provided det(P ) ̸= 0.
Proof. The result follows immediately from the definition of ϕ|UP .
Proposition 5.13. Let ϕ ∈J n,spezk,T and P ∈ Mn(O) be non-singular. Then ι(ϕ|UP )
is non-zero.
61
Proof. For this proof we write ιT and ι t for the Eichler Zagier maps of index TPTP
and tPTP respectively. In order to show ι t(ϕ|UP ) is non-zero we showPTP
ι t(ϕ|UP )(τ) = ιT (ϕ)(τ). (5.37)PTP
First we relate the theta coefficients of ϕ|UP to those of ϕ. Consider when s ̸∈
P (O#)n. In this case the sth theta coefficient of ϕ|UP is zero because for any N ∈
I t we have
PTP ,s
α tOn(N) = aϕ|U (m, r) = 0 (5.38)s+PTP P
as every Fourier coefficient of ϕ|UP with r ̸∈ P (O#)n is zero by Equation (5.36).
Now suppose s ∈ P (O#)n. Let Ps′ = s, and for N ∈ Z≥0 let f(N) := N| .det(P )|2
Using Lemma 4.5 directly reveals that f gives a well-defined bijection from I t
s,PTP
to Is′,T . From this and Equation (5.36) we see that hs(τ) = hs′(τ). Thus we see
that
( )
ι t(ϕ|UP ) = ιT (ϕ) | det(P )|2τ ≠ 0. (5.39)PTP
5.2 Twists of the Eichler--Zagier Map
In this section we’ll define twists of the Eichler--Zagier map. These slight variations
of the Eichler--Zagier map can be non-zero on a given input even when our original
Eichler--Zagier map is zero on that input. Though we are unable to generalize their
result here, Anamby and Das in [AD19, Proposition 3.2], were able to show that
for any non-zero form ϕ ∈ Jk,p, there exists some character such that the twist of
the Eichler--Zagier map by that character sends ϕ to something non-zero. We know
62
introduce these twists in oursetting. LetEn(D)d(T ) if D is oddf =  (5.40)En(D)d(T )/2 if D is even
Let ψ : Z → C be a character modulo 2fE (D)d(T ) and g : (O#)n/TOnn → C be
such that, for any d ∈ Z and s ∈ (O#)n/TOn we have
g(ds) = ψ(d)g(s). (5.41)
Define the twist of the Eichler--Zagier map by g to be
∑
ιg(ϕ) = g(s)hs(En(D)d(T )τ). (5.42)
n
s∈(O#) /TOn
One can show, in similar fashion to our proof for ι, that ιg(ϕ) ∈ Sk−n(Γ0(2fEn(D)d(T ), χDψ).
See [Hav95, Proposition 5.8]. We now introduce a group G and give an action of
this group on J . This action gives J the structure of a G-representation and hence
allows us to decompose the space into eigenspaces. We then see that these eigenspaces
interact nicely with our twisted Eichler--Zagier maps.
Definition 5.14. Let
G := {µ+ En(D)d(T ) ∈ O/En(D)d(T )O : N(µ) ≡ 1 (mod En(D)d(T ))} . (5.43)
Then G is a multiplicative group.
Proposition 5.15. Define, for µ ∈ G,
Wµ∑: J n nT,k(O) →∑JT,k(O) (5.44)
θshs →7 θshµs. (5.45)
Then {Wµ} is a commuting family of diagonalizable maps.
63
Proof. First we check that Wµ gives a well-defined map. We verify the follow three
conditions:
1. Wµ(ϕ) transforms properly with respect to ϵM ∈ U(1, 1).
2. Wµ(ϕ) transforms properly with respect to [λ, µ] ∈ On ×On.
3. Wµ(ϕ) to has an appropriate Fourier expansion.
By equations 5.13 and 5.8 we have 
1 1 [ ]hµs|k−n  = e −µstT−1µs hµs. (5.46)
0 1
After computing θs|T,nϵI directly from the definition and using the relationship
between the transformation for Θ and H, we can get that
hµs|k−nϵI = ϵhϵ−1µs. (5.47)
Finally we have
∑ [ ]
hµs|
1
k−nJ = √ e −2Re[Nm(µ)sT−1s′] hµs′ . (5.48)
det(i DT )
s′∈(O#)n/TOn
One can quickly check that the conditions on µ give
[ ] [ ]
e −2Re[Nm(µ)sT
−1s′] hµs′ = e −2Re[sT−1s′] hµs′ . (5.49)
 
Hence, since 1 1

, ϵI, J generate U(1, 1), we have
0 1 
Hµ|ϵM = UT (ϵM)Hµ. (5.50)
This, together with the transformation law for Θ, gives the transformation law
for Wµ(ϕ).
64
One can compute directly that θs|[µ, λ] = θs. This will give the desired transformation
for Wµ(ϕ).
We move onto the Fourier exp[a(nsion of Wµ(ϕ). Writin)g this out naiv]ely gives∑ N
α t −1µs+TOn(N)e + r T r τ + rw + r
tz . (5.51)
En(D)d(T )
(s,r,N)∈S1
Here
{ }
S = (s, r,N) ∈ (O#)n/TOn × (O#)n1 × Z≥0 : r ≡ s (mod TOn) and N ∈ IT,s .
(5.52)
If we define m(N, r) = N +rtT−1r then we have f(s,N, r) := (m(N, r), r)
En(D)d(T )
gives a well-defined bijection between S1 and S2 where
   m rt S +2 = (m, r) : ∈ Λn+1(O) . (5.53)r T
This is essentially the same as the argument that gives the existence of theta
expansions given in Proposition 4.1. From this the Fourier expansion will follow.
It is clear that this family of operators commutes so it remains to show that that
each operator is diagonalizable. By Proposition 5.6 we know that J nT,k is a finite
dimensional space. The fact that each Wµ is diagonalizable just follows from the
fact that G is a finite group and µ 7→ Wµ gives a representation of G on J nT,k.
Corollary 5.16. If, for a character η : G → C we define
{ }
J n,ηk,T := ϕ ∈ J
n
T,k : Wµ(ϕ) = η(µ)ϕ . (5.54)
then we have
⊕
J n = J n,ηT,k T,k . (5.55)
n,η
65
We now can study how how twists act on these spaces.
Proposition 5.17. Let g : (O#)n/TOn → C be such that g(µs) = η(µ)g(s) for
µ ∈ G. Recall that we have a map ι ng : JT,k → Sk−n(Γ0(2fEn(D)d(T ), χDη). For a
character η′ ≠ η we have ιg(Wµ′(h)) = 0.
Proof. We have
∑ ∑
ιg(ϕ) = g(s)hs(En(D)d(T )τ) = g(µs)hµs(En(D)d(T )τ). (5.56)
n
s∈(O#) /TOn s
Since Wµ(ϕ) = η(µ)ϕ we have, by the linear independence of theta functions, hµs =
η(µ)hs. We then have
∑
ιg(ϕ) = η(µ)η
′(µ) g(s)hs(En(D)d(T )τ) = η(mu)η
′(µ)ιg(ϕ). (5.57)
s
If we choose µ such that η(µ) ̸= η′(µ) then we see ιg(ϕ) = 0 as desired.
In the next section we explore applications of the main results presented in
Chapters 4 and 5.
66
CHAPTER 6
NON-VANISHING FOURIER COEFFICIENTS
6.1 Vector-valued Hermitian Modular Forms
The main goal of this chapter is to prove that an Hermitian modular form
has infinitely many non-zero Fourier Jacobi coefficients. We give an argument by
induction on the degree of the Hermitian modular form. The base case follows
from the fact that classical modular forms have infinitely many non-zero Fourier
coefficients. In order to use the induction hypothesis we need to work with vector-
valued Hermitian modular forms. We start by introducing the theory of those forms
here. For a reference on the basic theory see [FM15].
Definition 6.1. Fix a quadratic imaginary field K, positive integer n ≥ 1, a vector
space V and a representation ρ : GLn(C) × GLn(C) → GL(V ). Let U(n, n) =
U(n, n)(O). A vector-valued Hermitian modular form of weight ρ and degree n is a
holomorphic function F : Hn → V such that( )
t −1
 F (MZ) = ρ CZ +D,CZ +D F (Z) (6.1)
A B
for any  M =   ∈ U(n, n).
C D
We’ll be studying those with polynomial representations.
Definition 6.2. A representation ρ : GLn(C) → GL(V ) is polynomial if there is a
basis of V such that the coordinate functions ρij : GLn(C) → C are polynomial in
the entries of the input matrix. Similarly we can define the notion of a polynomial
representation on GLn(C)×GLn(C).
67
In order to use our induction hypothesis we need to link F to an Hermitian
modular form of smaller degree. Below we give the definition of this related, smaller
degree, Hermitian modular form. This follows a similar construction given by Böcherer
and Das in [BD22, Section 3.1.]. Note that the reduction in degree entails moving
to vector-valued Hermtian modular forms, even when starting with a scalar-valued
Hermitian modular form.
Definition6.3. Let F : Hn+1 → V be an Hermitian modular form. For A ∈ Hn+1
write τ wA =  with w ∈ Cn a row vector, z ∈ Cn a column vector and Z ∈ H .
z Z   n
τ w
For fixed the map  τ, Z (w, z) →7 F   is a holomorphic function on an
z Z
open set containing (0, 0).Hence wecan take a Taylor expansion to getτ w ∑ ′F = Fλ,λ′(τ, Z)wλzλ (6.2)
z Z λ,λ′
for some holomorphic functions Fλ,λ′ . Here λ and λ′ are multi-indices. For two such
multi-indices let ν(λ, λ′) be the degree, that is the sum of the entries in both tuples.
Choose the multi-index (λ, λ′) of the smallest degree such that Fλ,λ′ ̸= 0. If this
degree is ν0 then define
∑
λ′
F 0(τ, Z) := Fλ,λ′(τ, Z)x
λ2 λn 2
2 · · ·xn y2 · · ·
′
yλnn . (6.3)
(λ,λ′)
ν(λ,λ′)=ν0
We view F 0 as a function from H × Hn → C[x2, . . . , xn, y2, . . . , yn]ν0 . Here the
subscript indicates the polynomials are homogeneous of degree ν0.
Our goal is to show that, for fixed τ , F 0(τ, Z) is a vector valued Hermitian
modular form of degree n. Then we’ll show that the Fourier coefficients of F 0 are
68
the Fourier-Jacobi coefficients of F . F 0 will have infinitely many non-zero Fourier
coefficients by induction and we’ll be able to conclude that F 0 has infinitely many
non-zero Fourier-Jacobi coefficients as desired.
The main difficulty in showing that F 0 is a vector valued Hermitian modular
form of degree n is showing that F 0 satisfies the proper transformation law. The
following lemma will be necessary. We first introduce some useful notation. Let
A Bg =  ∈ U(n, n) and Z ∈ Hn. Then we define
C D
 1 0 0 0

ĝ :=

0 A 0 B

(6.4)
0 0 1 0
0 C 0 D
and
µg(Z) :=CZ +D and λg(Z) := CZt +D. (6.5)
A B
Lemma 6.4. Let  g =  ∈ U(n, n). We have
 C D   
ĝ τ w τ − w(CZ +D)−1Cz w(CZ +D)−1=  (6.6)
z Z λ (Z)−tg z gZ
Proof.Direct c 
omputation gives −1  
0 0τ w 1 0+   1 0 =  (6.7)
0 C z Z 0 D −(µ (Z))−1Cz (µ (Z))−1g g .
One can then com 
pute
1 0
       −1
τ w 0 00 0τ w 1 0+ + 
0 A z Z 0 B 0 C z Z 0 D
69
 τ − w(µg(Z))−1cz −w(µg(Z))−1= 
Az − (gZ)Cz gZ
To prove the claim it suffices to prove that
(λg(Z))
−t = A− (gZ)C. (6.8)
We know A,B,C and D satisfy
t − tD A B C = In (6.9)
t
A C − tC A = 0 (6.10)
  BD −DB = 0. (6.11)
because A B ∈ U(n, n). From these equations it follows that
C D
t t
λ tg(Z) A = ZA C + In +B C
t t
λg(Z)
t(AZ +B) = (ZA +B )µg(Z)
from which Equation (6.8) follows.
Definition 6.5. Let F : Hn+1 → V be a Hermitian modular form of weight
ρ : GLn+1(C)×GLn+1(C) → GL(V ). (6.12)
Define a representation
ρ0 : GLn(C)×GLn(C) → GL(V ⊗ C[x1, . . . , xn, y1, . . . , yn]v0) (6.13)
by
ρ0(α, β) · v ⊗ f(x⃗, y⃗) := (ρ(α̂, β̂) · v)⊗ f(xα, βTy). (6.14)
70
Proposition 6.6. For fixed τ ,
F 0(τ, Z) : Hn → V ⊗ C[x1, . . . , xn, y1, . . . , yn]v0 (6.15)
is a Hermitian modular form of weight ρ0.
Proof. The fact that F 0 is holomorphic follows immediately from the holomorphy
of each Fλ,λ′ which in turn follows from the holomorphy of F . Next we consider the
τ w
transformation law. Let Z  = ∈ Hn+1. Lemma 6.4 implies
z Z
∑
F | ′ρĝ = ρ(µ −1 −1 −t λ −1 λĝ(Z), λĝ(Z)) · Fλ,λ′(τ − w(µg(Z)) Cz, gZ)((λg(Z)) z) (w(µg(Z)) ) .
λ,λ′
(6.16)
If we consider the Taylor expansion of Fλ,λ′ we see∑ ′
F |ρĝ = ρ(µ −1 −t λ −1 λĝ(Z), λĝ(Z)) · Fλ,λ′(τ, gZ)((λg(Z)) z) (w(µg(Z)) ) + h.o.t.
λ,λ′
ν(λ,λ′)=ν0
(6.17)
It follows that
F 0(τ, Z) = ρ0(CZ +D,CZT +D)−1F 0(τ, gZ). (6.18)
6.2 Vector-Valued Hermitian Jacobi Forms
With vector-valued Hermitian modular forms we can construct vector-valued
Hermitian Jacobi forms similarly to the scalar case. Rather than give a general
theory of such forms we do the bare minimum to relate these forms to the scalar
setting so that we can use the previously developed theory there.
71
Definition 6.7. Let V be a finite dimensional complex vector space and ρ : GLn+1(C)×
GLn+1(C) → GL(V ) be a representation. A vector-valued Hermitian Jacobi form of
degree n, index T and weight ρ is a holomorphic function φ : H × Cn × Cn → V
such that
 a b1. For all M = ϵ  ∈ U(1, 1)
c d


   −1
ϵcτ + ϵd ϵcw ϵcτ + ϵd ϵcztφ(τ, w, z) = ρ ,  e (−cwTz/j(M, τ))φ (ϵM · (τ, w, z))
0 In−1 0 In−1
(6.19)
2. For all λ, µ ∈ OnK  1 −λ  

1 −λ 
−1 t T tφ(τ, w, z) = ρ , eT (λ λτ + zλ+ λ w)φ(τ, w + λτ + µ, z + µt + λ τ).
0 In−1 0 In−1
(6.20)
3. φ has a Fourier expansion of the form
∑
φ(τ, w, z) := α(m, r)e(mτ + wr + rz) (6.21)
(m,r)∈ST
where α(m, r) ∈ V .
Proposition 6.8. Let F : Hn+1 → V be a vector valued Hermitian modular form of
weight ρ. Let F have Fourier expansion
∑
F (Z) = a(F,A)e(AZ). (6.22)
A∈Λn+1(O)
72
For T ∈ Λn(O) define
∑ 
  m rt
ϕT (τ, w, z) := a F,  e(aτ + wr + rtz). (6.23)
m∈Z,r∈(O#)n r T

 
 t
m r ∈Λ+n+1(O)
r T
Then


τ w ∑F = φT (τ, w, z)e(TZ) (6.24)
z Z T∈Λn(O)
and each φT is a vector valued Hermitian Jacobi form of degree n− 1, weight ρ and
index T .
Proof. See the proof of Proposition 3.3. This follows in a nearly identical fashion.
Our next goal is to show that if φ is a vector valued Hermitian Jacobi form
then it has a component which is a scalar valued Hermitian Jacobi form. This will
allow us to use our already developed theory in this setting.
Proposition 6.9. Let φ be a non-zero Hermitian Jacobi form of degree n, index T
and weight ρ : GLn+1(C) × GLn+1(C) → GL(V ). Suppose that ρ is a polynomial
representation. Then there exists a basis for V and a component of φ with respect
to this basis that is a vector-valued Hermitian Jacobi form with co-domain C.
Proof. First recall the Lie-Kolchin theorem: For a connected solvable linear algebraic
group G and a representation ρ : G → GL(V ), the image ρ(G) is simultaneously
triangularizable. Let Bn(C) ⊂ GLn(C) be the subgroup of upper triangular matrices.
Recall that Bn(C) is both solvable and connected. Thus Bn(C) × Bn(C) is as well
73
and so we can apply the Lie-Kolchin theorem to get a basis {v mi}i=1for V such that,
whenever A,B are upper triangular ρ(A,B) is upper triangular in this basis.
Let φi be φ composed with the ith coordinate function from V to C with respect
to this basis and let r be the largest i such that φi is non-zero. I claim that
φr : H× Cn × Cn → C (6.25)
is a scalar valued Hermitian Jacobi form. First we determine the weight. Let ρr,r(A,B)
be the (r, r) entry of ρ(A,B). I claim that the map 
s 
z 0  
f1 : z →7 ρr,r   , In
0 In−1
7→ 
  
f2 : z ρr,r  z 0 In, 
0 In−1
are polynomial homomorphisms from C to C. If a representation is polynomial in
one choice of basis, then it is polynomial in any choice of basis. Hence f1, f2 ∈
C[z]. When ρr,r is restricted to upper-triangular matrices it is a homomorphism
because on the ring of upper-triangular matrices the (r, r)th coordinate map is a
homomorphism. Thus f1 and f2 are polynomial homomorphisms. The only such
maps are of the form z 7→ λk. Let fi(z) = zki . Define a one-dimensional representation
of GLn(C)×GLn(C) by
(A,B) 7→ det(A)k1 × det(B)k2 . (6.26)
I claim this is the weight of φr.
a b
Let  M = ϵ  ∈ U(1, 1). First we consider transformation (1). We have
c d
   ϵcτ + ϵd ϵcw ϵcτ + ϵd ϵcwt,  (6.27)
0 In−1 0 In−1
74
      1 ϵcw 1 ϵcwtϵcτ + ϵd 0  ϵcτ + ϵd 0 = , ,  (6.28)
0 In−1 0 In−1 0 In−1 0 In−1
If (A,B) are unitriangular, that is they are upper triangular with 1’s along the
diagonal, then so is ρ(A,B) since the subgroup of unitriangular matricesis the
1 ϵcw 1 ϵcwt
derived subgroup of the Borel subgroup. Thus    ρ  ,  will be
0 In−1 0 In−1
unitriangular and will leave the rth component of a vector unchanged. We have
then, using the transformation lawforφ, −1ϵcτ + ϵd 0  ϵcτ + ϵd 0 φr(τ, w, z) = ρr,r ,  e (−cwTz/j(M, τ))φr (ϵM · (τ, w, z))
0 In−1 0 In−1
= (cτ + d)−k1−k2ϵk2−k1e (−cwTz/j(M, τ))φr (ϵM · (τ, w, z))
as desired.
Next we check the other desired transformation. We have, for φ
1 −λ  1 −λ 
−1
φ(τ, w, z) = ρ ,  [ ]t te λTλ τ + wTλ + λTz (6.29)
0 In−1 0 In−1
· tφ(τ, w + λτ + µ, z + λ τ + µ). (6.30)
Since φ takes uni-triangular touni-triangular we see, 
for λ ∈ On
1 −λ  1 −λρ ,  (6.31)
0 In−1 0 In−1
is uni-triangular. A comparison of the rth entries on either side and the fact that
φr is the last non-zero entry in φ gives the desired result.
The existence of the proper Fourier series for φr follows directly from the Fourier
series for φ and the definition of φr.
75
Remark 6.10. A vector-valued Hermitian Jacobi form of scalar weight, as in the
result above, and a Hermitian Jacobi form are not quite the same. The only difference
being transformation with respect to the matrices ϵM ∈ U(1, 1) when ϵ ̸= ±1. That
said, the important theory we have developed, i.e. that of theta expansions and
theta coefficients, will still exist for vector-valued Hermitian Jacobi forms of scalar
weight.
6.3 Non-zero Fourier Coefficients
Now that we’ve linked vector-valued Hermitian Jacobi forms to the scalar setting
we can prove a few results on the non-zero Fourier coefficients of Hermitian modular
forms.
Proposition 6.11. Let F be a non-zero vector valued Hermitian modular form of
degree n ≥ 2. Then F has infinitely many non-zero Fourier Jacobi coefficients with
non-singular index.
Proof. We prove the result by induction on the degree n. First consider when n =
2. Let F 0 be as in Definition 6.3 and let V be the codomain of F 0 and fix τ0 so
that F 0(τ0, Z) is non-zero as a function of Z. By Proposition 6.6 we know F 0 is an
Hermitian modular form. Analogously to Proposition 6.9 we can choose a basis for
V such that F 0 has a non-zero component, say F 0r that is a scalar-valued modular
form. Because F 0r has infinitely many Fourier coefficients with non-zero index so
too will F 0. If F has Fourier Jac∑obi expansion
φn(τ, w, z)e(nZ) (6.32)
n≥0
then ∑ ∂ ∂
aF 0(n) = c(τ0) ′φn(τ0, w, z)|w,z=0 (6.33)λ λ
ν(λ,λ′
∂w ∂z
)=ν0
76
for some constant c(τ0). Hence, for each n > 0 with aF 0(n) ≠ 0 we must also have
φn ̸= 0. Then F has infinitely many non-zero Fourier-Jacobi coefficients with non-
singular index because F 0 has infinitely many non-zero Fourier coefficients aF 0(n)
with n > 0.
Now suppose we know the result for vector valued Hermitian forms of degree n and
let F have degree n + 1. Let F 0 be as in Definition 6.3 and again call the codomain
V . By hypothesis F 0 has infinitely many non-zero Fourier Jacobi coefficients of
non-singular index. Let φ be one such coefficient say of index T . By Proposition
6.9 we can choose a basis of V and a component of φ with respect to this basis
such that φ nr : H× C × Cn → C is a vector valued Hermitian Jacobi form of scalar
weight. As in Remark 6.10 φr will have a theta expansion. Choose a non-zero theta
coefficient hs. By Corollary 5.4 hs will be a classical modular form and hence will
have infinitely many non-zero Fourier coefficients of non-zero index. Each such
coefficient of hs will give rise to a non-zero Fourier coefficient of φr with non-singular
index. Recall if αs+TOn(N) ̸= 0 thenφr hasaFourier coefficient α(m, r) ≠ 0 with
m rtd  = N. (6.34)
r T
This Fourier coefficient of non-singular index for φr will guarantee one for F 0. An
argument identical to the one given at the end of the preceding paragraph will
show that for each Fourier coefficient of hs with non-zero index, we get a different
non-zero Fourier Jacobi coefficient of F with non-singular index. Thus, since hs has
infinitely many non-zero Fourier coefficients with non-zero index, F has infinitely
many non-zero Fourier Jacobi coefficients with non-singular index.
Next we’ll look at the actual Fourier coefficients of F . This result reduces to
the Sturm bound for classical modular forms though in dimensions ≥ 2 it is not
77
nearly as restrictive as that result. We no longer need consider vector valued Hermitian
∑Jacobi forms. First recall the Sturm bound which states the following: Let f =
anq
n be a classical modular form of weight k and level N . If an = 0 for all
n ≤ ⌊km⌋ then f = 0. Here
12 ∏( )1
m = [SL2(Z) : Γ0(N)] = N 1 + . (6.35)
p
p|N
We now state our generalization of this result.
Definition 6.12. Define, for k ≥ 1, n ≥ 1 and D ∈ R,
n
(k − n)D3+2⌊ 2 ⌋
Ck,n,D = . (6.36)
12
Proposition 6.13. Let ϕ be a non-zero Hermitian Jacobi form of non-singular
index T , weight k and degree n. Suppose that ϕ has Fourier expansion
∑ [ ]
α(m, r)e mτ + wr + rtz . (6.37)
(m,r)∈ST
Then there exists (m, r) ∈ STwith α(m, r) ≠ 0 andm rt

det  ≤ C 2k,n,D det(T ) . (6.38)
r T
Proof. We start with the theta expansion
∑
ϕ = θshs. (6.39)
n
( )s∈(O#) /TOn
Since ϕ ̸ n= 0 we can find s ∈ O# /TOn such that hs ̸= 0. By Corollary 5.4
we have hs ∈ Sk−n (Γ(En(D)d(T ))). The Sturm bound for classical modular forms
implies that we can find a non-zero Four(ier coeffi)cient of hs, say αs+TOn(N) with
(k − n)(En(D)d(T ))3 ∏ − 1 ≤ (k − n)(En(D)d(T ))3N < 1 . (6.40)
12 p2 12
p|En(D)d(T )
78
Wesee that ϕ will have a non-zero Fourier coefficient with index (m, r) such that
m rtd  = N and r ≡ s (mod TOn) by definition of hs. From this the result
r T
follows.
Proposition 6.14. Let F be an Hermitian modular form of weight k and degree
n ≥ 2 with non-zero T th Fourier Jacobi coefficient for some non-singular T . Then
F has a non-zero Fourier coefficient with indexA ∈ Λn(O) such thatm rtA =  (6.41)
r T
and
det(A) ≤ Ck,n−1,D det(T )2 (6.42)
Proof. Let ϕT be the non-zero T th Fourier Jacobi coefficient of F . Then ϕT is a
non-zero Hermitian Jacobi form of weight k, index T and degree n−1. By Proposition
6.13 ϕT has a non-zero Fourier coefficient α(m, r) with
detm rt ≤ C 2k,n−1,D det(T ) . (6.43)
r T
Since  
am rt = α(m, r) ̸= 0 (6.44)
r T
by definitionof φT (see Proposition 3.3.) we see that the proposition is satisfied for
m rtA = .
r T
.
79
Corollary 6.15. Let F be a non-zero Hermitian modular form of weight k and
degree n ≥ 2. Then F has infinitely many non-zero Fourier coefficients with index
m rtA =  such that
r T
det(A) ≤ C 2k,n−1,D det(T ) (6.45)
Proof. By Proposition 6.11 we know F has infinitely many non-zero Fourier Jacobi
coefficients whose index is a non-singular matrix. For each such coefficient we will
get a non-zero Fourier coefficient with index A satisfying the conditions of the proposition.
In this chapter we’ve seen a few consequences the developed theory has on the
theory of Hermitian modular forms. As stated in the introduction original intent
of this work was to generalize some results on non-vanishing Fourier coefficients of
Hermitian modular forms to higher degree. In the next chapter we discuss some of
the issues in generalizing their results to this setting.
80
CHAPTER 7
FUNDAMENTAL FOURIER COEFFICIENTS
7.1 Difficulties and Roadblocks
The goal of this final chapter is to explain several difficulties in using this theory
to generalize the papers of Böcherer and Das [BD22] and Anamby and Das [AD19].
Though the approaches of Böcherer and Das are superficially distinct, the fundamental
issue to generalizing both is the same. First we outline the general approach of
Böcherer and Das. The main theorem of their paper is as follows:
Theorem 7.1. Let F be a non-zero vector valued Seigel modular form of weight ρ
and degree n. Suppose further that k(ρ)− n ≥ ϱ(n). When n is even, assume that F
2
is cuspidal. Then there exists infinitely many GLn(Z) inequivalent matrices T ∈ Λ+n
such that d(T ) is odd and square free, and aF (T ) ̸= 0.
Note here two matrices in T, T ′ ∈ Λ+n are inequivalent over GLn(Z) if
t
A TA ̸=
T ′ for any A ∈ GLn(Z). The overarching strategy is a proof by induction on the
degree n of the form. The base case of n = 1, which is the above statement for
classical modular forms, is proven in [AD19].
The first step in the induction is to construct a non-zero Fourier Jacobi coefficient
φT for which T has an odd, square-free discriminant. This follows from the construction
of vector valued form F 0, very similar to that given in Definition 6.3, and the induction
hypothesis. From this φT they derive infinitely many non-zero Fourier coefficients
aF (A) such that dA is odd and square free. Böcherer and Das then prove, in analogue
to 6.9, that φT has a non-zero component which is a scalar-valued Jacobi form, say
(r)
φT .
The final step before dealing with classical modular forms is to construct a non-
81
zero theta coefficient of this scalar-valued Jacobi, say hµ, such that µ is primitive.
Here µ ∈ Zn−1/2TZn−1 is primitive if
µ t
T−1
µ
(7.1)
2 2
has the largest possible denominator. This step is accomplished in [BD22, Proposition 3.5]
and is the step that I was unable to generalize to my setting.
The existence of such a primitive hµ together with the base case and the relationship
between the Fourier coefficients of F and those of hµ prove the result.
We say a few words about generalizing this approach to the Hermitian setting. As
illustrated in Chapter 6 we can generalize these results and constructions until
we need the existence of a primitive hµ. In what follows we give a definition for
primitivity in our setting and explain the particular difficulty we faced in proving
Proposition 3.5. from Böcherer and Das’ paper in this setting.
( )
Definition 7.2. We say µ ∈ O# n is primitive with respect to T ∈ Λ+(O) if
1 ( √ )
n
−1 √
√ i DT [i Dµ]
i D
√
has denominator exactly d(T )T D if n is odd and d(T ) if n is even. Similarly µ ∈
(O)n is primitive if
1 ( √ )−1√ i DT [µ]
i D
√
has denominator exactly id(T ) D if n is odd and d(T ) if n is even. This notion of
√
primitive descends to On/i DOn which can be seen by computing
1 √ √
( ) √( (i DT )
−1[µ+ i DTq] (7.2)
i D
1 √ −1√ √1
√ )
= i DT [µ] + µtq − qtµ− qt(i DT )q . (7.3)
i D i D
82
The existence of a non-zero primitive theta coefficient in Böcherer and Das
boils down to proving that the m[ atrix( )]
e µν (7.4)
p µ∈Z/pZx
ν∈Z/pZ, ν2=ν20
has maximal rank for some fixed ν0 ∈ Z/pZ. Such a matrix either has 2 columns
or one column depending on whether ν0 ≡ 0 or not and so demonstrating that this
matrix has maximal rank is relatively straightforward.
In the Hermitian setti[ng(the analogou)s]matrix looks like
(µν + νµ)
e (7.5)
p µ∈O/pO×
ν∈O/pO, |ν|2≡|ν0|2 (mod p)
and in particular the number of columns is equal to the number of solutions
to |ν|2 ≡ |ν |20 in O/pO, which could be as high as 2p − 1 if p is split in O. The
matrices that appear here can fail to have maximal rank and hence are not sufficient
to prove that φT has a non-zero primitive theta coefficient. Without such a coefficient
we are still able to lift a Fourier coefficient of hµ to that of F , we just can no longer
guarantee that the index of this Fourier coefficient of F will be primitive. In [AD19]
Anamby and Das give the following result:
Theorem 7.3. Let F be an Hermitian cusp form of degree 2. Then a(F, T ) ̸= 0
for infinitely many matrices T such that D det(T ) is of the form pαKn where n is
square-free with (n, pK) = 1 and 0 ≤ α ≤ 2 if D ̸= 8 and 0 ≤ α ≤ 2 if D ̸= −8 and
0 ≤ α ≤ 3 if D = −8.
Our goal would be to generalize this result beyond degree 2. To prove this
result Anamby and Das essentially leverage the Eichler--Zagier map and the twists
there-of to relate the Fourier coefficients of a non-zero Fourier Jacobi coefficient
to those of a classical modular form. By conjugating by a matrix in GL2(O), they
83
are able to guarantee the existence of a non-zero Fourier Jacobi coefficient of prime
index. The central result of their work is that, for forms of prime index, either the
Eichler--Zagier map or a twist of this map, is injective. This allows them to get a
non-zero classical cusp form of a certain index and level with Fourier coefficients
equal to those of the Hermitian Jacobi form. Then, having reduced to the case of
classical modular forms, they prove a non-vanishing result in this setting.
The roadblock in trying to generalize Anamby and Das is essentially the same.
The first is getting a Fourier Jacobi of prime determinant. Anamby and Das use a
result specific to the setting of n = 2, though this can possibly be circumvented
by using an induction argument. The second, and more fundamental, is trying to
generalize the non-vanishing of either the Eichler--Zagier map or a twist on a given
form of prime index.
To prove this result Anamby and Das make the following argument. If each twist
sends a form to zero, then all the "primitive" theta coefficients must be zero. They
then, just like in Böcherer and Das, look at hs|J , get a family of sums that must
be zero and from this construct a matrix they argue must be of maximal rank and
arrive at a contradiction with the fact that all the "primitive" theta coefficients are
zero. Thus we run into the same issue.
84
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