APPROVED: Olav K. Richter, Major Professor William Cherry, Committee Member Charles H. Conley, Committee Member Su Gao, Chair of the Department of

Mathematics Mark Wardell, Dean of the Toulouse

Graduate School

HERMITIAN JACOBI FORMS AND CONGRUENCES

Jayantha Senadheera

Dissertation Prepared for the Degree of

DOCTOR OF PHILOSOPHY

UNIVERSITY OF NORTH TEXAS

August 2014

Senadheera, Jayantha. Hermitian Jacobi Forms and Congruences. Doctor of

Philosophy (Mathematics), August 2014, 60 pp., 30 numbered references.

In this thesis, we introduce a new space of Hermitian Jacobi forms, and we

determine its structure. As an application, we study heat cycles of Hermitian Jacobi

forms, and we establish a criterion for the existence of U(p) congruences of Hermitian

Jacobi forms. We demonstrate that criterion with some explicit examples. Finally, in

the appendix we give tables of Fourier series coefficients of several Hermitian Jacobi

forms.

ii

Copyright 2014

by

Jayantha Senadheera

iii

ACKNOWLEDGEMENTS

First and foremost, I would like to thank my thesis advisor, Dr. Olav Richter, for

introducing me to the wonderful area of automorphic forms. I highly appreciate his

excellent guidance, understanding, and patience throughout my work on automorphic

foms. I am extremely grateful to the faculty and staff of the mathematics department at

the University of North Texas for arranging a great research environment to carry out

my studies.

I take this opportunity to express my special thanks to Dr. Martin Raum, for

providing SAGE codes to compute Hermitian Jacobi forms, which were extremely

helpful at the initial stages of this research. I greatly appreciate the support of my

friends Dhanyu and Fazeen, who assisted whenever I needed help with computer related

issues.

I will forever be thankful to Dr. W. Ramasinghe for encouraging me to return to

mathematics again, and for offering me an academic position, even 10 years after I had

received a bachelor's degree. I will also be forever thankful to Dr. Sunil Gunarathne for

encouraging me to maintain my interest in number theory. Both of them gave me

enormous support for pursuing a Ph.D. in the USA.

I am extremely grateful to my beloved wife Janakee and my beloved son

Matheesha for their unconditional support, patience, and encouragement during my

studies.

Finally, I dedicate my dissertation work to my loving parents, the late Mrs.

Wimala Mahathanthile and Mr. Adwin Senadheera with a special feeling of gratitude.

iv

TABLE OF CONTENTS

Page ACKNOWLEDGMENTS ............................................................................................... iii CHAPTER 1. INTRODUCTION .................................................................................... 1 CHAPTER 2. MODULAR FORMS ................................................................................ 3

2.1 Definition and Examples ............................................................................ 3 2.2 U(p) Congruences of Modular Forms ......................................................... 6

CHAPTER 3. CLASSICAL JACOBI FORMS ................................................................ 8

3.1 Definition of Jacobi Group and Jacobi Forms ............................................ 8 3.2 Taylor Development of Jacobi Forms and the Structure Theorem for Jk,1

.................................................................................................................. 10 3.3 The Theta Decomposition ......................................................................... 12

CHAPTER 4. HERMITIAN JACOBI FORMS ............................................................. 14

4.1 The Hermitian Jacobi Group and Hermitian Jacobi Forms ...................... 14 4.2 The Theta Decomposition ......................................................................... 17 4.3 The Structure of Hermitian Jacobi Forms of Index 1 ............................... 22

CHAPTER 5. U(p) CONGRUENCES OF HERMITIAN JACOBI FORMS ................. 29

5.1 Congruences and Filtrations ..................................................................... 29 5.2 Examples ................................................................................................... 37

APPENDIX. FOURIER COEFFICIENTS OF HERMITIAN JACOBI FORMS .......... 41 BIBLIOGRAPHY ........................................................................................................... 58

CHAPTER 1

INTRODUCTION

Eichler and Zagier [7] systematically developed a theory of Jacobi forms, which are

holomorphic functions of two complex variables that satisfy certain transformation laws

under the action of the Jacobi group. Jacobi forms appear naturally in different areas of

mathematics and physics. In particular, they occur as Fourier-Jacobi coefficients of Siegel

modular forms of degree 2. This link played an important role in the solution of the Saito-

Kurokawa conjecture (see [15, 16, 17, 2, 30]).

Hermitian modular forms are generalizations of Siegel modular forms, and Hermitian

Jacobi forms are holomorphic functions of three complex variables that occur as Fourier-

Jacobi coefficients of Hermitian modular forms of degree 2. First Haverkamp [9, 10], and

later Sasaki [25] and Das [4, 5] studied Hermitian Jacobi forms over the Gaussian number

field Q(i), and they established several properties of such Jacobi forms. The usual heat

operator is an important tool in the study of classical Jacobi forms, which for example

allows one to explore congruences and filtrations of Jacobi forms (see [23, 24]). As in the

case of classical Jacobi forms, there is also a heat operator in the context of Hermitian Jacobi

forms (see (12)). However, the action of that heat operator on the Hermitian Jacobi forms

in [9, 25, 4, 5] is not natural, i.e., it cannot be “corrected” as in the case of classical Jacobi

forms. Thus, one needs a different notion of Hermitian Jacobi forms.

In this thesis, I introduce a more general definition of Hermitian Jacobi forms (see

Definition 4.3 and Definition 4.5), which permits the desired action of the heat operator. I

determine the structure of such forms if the index is 1. More precisely, I have shown (see

Theorem 4.14) that the ring of Hermitian Jacobi forms of index 1 over Q(i) is a free module

of rank 4 over the ring of elliptic modular forms.

As an application, I study “heat cycles” of Hermitian Jacobi forms of index 1, and

I establish a criterion for the existence of U(p) congruences of Hermitian Jacobi forms (see

Theorem 5.11). I present several explicit examples to illustrate Theorem 5.11. Recall that

1

U(p) congruences of elliptic modular forms have applications in the context of traces of

singular moduli and class equations (see Ahlgren and Ono [1], Elkies, Ono, and Yang [8],

and Chapter 7 of Ono [20]). It would be interesting to see if U(p) congruences for Hermitian

Jacobi forms also find further applications.

This thesis is organized as follows. In Chapter 2, I briefly recall the notion of a

modular form. In Chapter 3, I review the classical Jacobi forms of [7], and I state some of

their basic properties. In Chapter 4, I give a new definition of Hermitian Jacobi forms over

Q(i), and I prove Theorem 4.14. In Chapter 5, I discuss U(p) congruences of Hermitian

Jacobi forms over Q(i) of index 1, and I prove Theorem 5.11. Finally, in the Appendix , I

give tables of Fourier series coefficients of several Hermitian Jacobi forms.

2

CHAPTER 2

MODULAR FORMS

This brief chapter is on modular forms. In Section 2.1 I recall the definition and basic

examples of modular forms, and in Section 2.2 I review the notion of U(p) congruences for

modular forms. The content of Section 2.1 can by found in any text on modular forms, and

Ono [20] is a good reference for both sections. Throughout this chapter, k is a nonnegative

integer.

2.1. Definition and Examples

Let H be the usual complex upper half plane, τ ∈ H be a typical variable, q := e2πiτ ,

and

Γ := SL2(Z) ={

( a bc d )∣∣a, b, c, d ∈ Z, ad− bc = 1}

be the modular group. It is well known that Γ acts on H by linear fractional transformations,

and modular forms are equivariant with respect to this action.

Definition 2.1. A function f : H→ C is a holomorphic modular form of weight k on Γ if

it satisfies the following conditions:

(i) f is holomorphic on the upper half plane H.

(ii) f

(aτ + b

cτ + d

)= (cτ + d)kf(τ) ∀ ( a bc d ) ∈ Γ.

(iii) The Fourier series expansion of f is of the form

f(τ) =∞∑n=0

c(n)e2πinτ .

If c(0) = 0, then f is called a cusp form. Let Mk denote the space of holomorphic modular

forms of weight k and let Sk denote its subspace of cusp forms of weight k.

The definition of modular forms can easily be extended to other groups and also to

the case where k is a half-integer (see for example Ono [20]). In this thesis, I mostly deal

with integer weight modular forms for the full modular group, but in Chapter 3 I briefly

3

encounter modular forms of half-integral weight for the congruence subgroup

Γ0(4) :={

( a bc d ) ∈ Γ | c ≡ 0 (mod 4)}.

Next I recall Eisenstein series, which are essential examples of modular forms.

Example 2.2. Let k ≥ 2. The Eisenstein series of weight k are defined by

Ek(τ) := 1−2k

Bk

∞∑n=1

σk−1(n)qn,

where

σk−1(n) :=∑d|n

dk−1

is the usual divisor function, and where the Bernoulli numbers Bk are defined by

x

ex − 1=∞∑k=0

Bkxk

k!.

If k ≥ 4, then Ek ∈Mk. If k = 2, then E2 is a so-called quasimodular form, which satisfies

(1) E2

(aτ + b

cτ + d

)= (cτ + d)2 E2(τ) +

6c

πi(cτ + d) ∀ ( a bc d ) ∈ Γ .

Finally, any modular form on Γ can be expressed in terms of the Eisenstein series E4 and

E6. For example, the famous Ramanujan-Delta function can be written as

∆ :=1

1728(E34 − E26) ∈ S12.

Other important examples of modular forms are theta functions. I now recall the

classical Jacobi theta functions.

Definition 2.3. Let a, b ∈ R. Then

(2) θa,b(τ, z) =∑n∈Z

eπi(a+n)2τ+2πi(n+a)(z+b).

Of particular interest are the following special cases:

(i) θ0,0(τ, z) =∑eπin

2τ+2πinz

(ii) θ0, 12(τ, z) =

∑eπin

2τ+2πin(z+ 12

)

(iii) θ 12,0(τ, z) =

∑eπi(n+

12

)2τ+2πi(n+ 12

)z

4

(iv) θ 12, 12(τ, z) =

∑eπi(n+

12

)2τ+2πi(n+ 12

)(z+ 12

).

Note that some authors write these theta functions also as θ0,0(τ, z), θ0,1(τ, z), θ1,0(τ, z), and

θ1,1(τ, z). Specializing to z = 0 yields the classical theta constants:

x := θ0,0(τ, 0) = 1 + 2∞∑n=1

qn2

2

y := θ0, 12(τ, 0) = 1 + 2

∞∑n=1

(−1)nqn2

2

z := θ 12,0(τ, 0) = 2q

18

∞∑n=0

qn(n+1)

2 .

(3)

It is well known (see for example, Igusa [11]) that the following identities hold, which

imply that every holomorphic modular form on Γ can be expressed in terms of theta con-

stants.

(i) x4 = y4 + z4

(ii) E4 =12(x8 + y8 + z8)

(iii) E6 =12(x4 + y4)(x4 + z4)(y4 − z4)

(iv) ∆ = 2−8(xyz)8.

One can show that Mk and Sk are finite dimensional vector spaces, and I end this

section with their dimension formulas.

Theorem 2.4. Let bxc denote the greatest integer less than or equal to x. Then

dimMk =

0 if k < 0 or if k is odd,⌊k12

⌋if k ≥ 0 k ≡ 2 (mod 12),⌊

k12

⌋+ 1 otherwise.

In particular, if f ∈MK is nonconstant, then k ≥ 4 and k is even. Finally, if k ≥ 4, then

dimSk = dimMk − 1.

5

2.2. U(p) Congruences of Modular Forms

Let f(τ) =∑∞

n=0 a(n)qn be a modular form, and let d be a positive integer. Then

Atkin’s U -operator is defined by

f∣∣U(d) := ∞∑

n=0

a(nd)qn.

This operator is an important tool in the theory of modular forms. Of particular interest is

the case when d = p is a prime and when a(n) ∈ Z, and one is interested in the question for

which primes p is f∣∣U(p) ≡ 0 (mod p).

Example 2.5. Let ∆ ∈ S12 be the Ramanujan Delta function. Then ∆∣∣U(p) ≡ 0 (mod p)

if:

p = 2, 3, 5, 7 (Ramanujan [21] and Mordell [18])

p = 2411 (Newman [19])

p = 7758337633 (Lygeros and Rozier [14])

and there are no further p < 1010. It is not known if there a infinitely many primes p such

that ∆∣∣U(p) ≡ 0 (mod p).The Ramanujan theta operator Θ := q d

dq= 1

2πiddτ

acts on the Fourier series of f by

Θ( ∞∑n=0

a(n)qn)

=∞∑n=0

na(n)qn.

Remark 2.6. If f ∈Mk, then

(4) Θ(f) =k

12fE2 + f̂ ,

where f̂ ∈ Mk+2. In particular, if f = E4 or f = E6, then one finds Ramanujan’s [21]

identities

Θ(E4) =1

3(E4E2 − E6) and Θ(E6) =

1

2(E6E2 − E8).

Let p be a prime. It is easy to see that

Θp−1(f) ≡ f (mod p) ⇐⇒ f |U(p) ≡ 0 (mod p),

6

which yields a so-called theta cycle. Tate (see §7 of [12]) has initiated the theory of such

theta cycles, which is based on studying filtrations of modular forms. More precisely, let

M̃k :={f (mod p) : f(τ) ∈Mk ∩ Z[[q]]

},

and let

ω(f) := inf{k : f (mod p) ∈ M̃k

}denote the filtration of f modulo p. Tate’s theory of theta cycles yields the following criterion

for the existence of U(p)-congruences.

Theorem 2.7. Let f ∈Mk ∩ Z[[q]] with ω(f) = k. If p > k, then

ω(

Θp−k+1(f))

=

2p− k + 2, if f

∣∣U(p) 6≡ 0 (mod p);p− k + 3, if f

∣∣U(p) ≡ 0 (mod p).In Chapter 5, I will extend Theorem 2.7 to the case of Hermitian Jacobi forms of

index 1, and I will present several examples to my criterion.

7

CHAPTER 3

CLASSICAL JACOBI FORMS

In this chapter, I review some basic properties of Jacobi forms. All results in this

chapter (and many more details) are contained in [7]. Throughout this chapter, k and m are

nonnegative integers.

3.1. Definition of Jacobi Group and Jacobi Forms

In this section, I give the definition of classical holomorphic Jacobi forms and I recall

some concrete examples of holomorphic Jacobi forms.

First I define the Jacobi group. As in the previous Chapter, Γ = SL2(Z).

Definition 3.1. The set ΓJ := Γ n Z2 ={

(M,X) |M ∈ Γ, X ∈ Z2}

forms a group under

the group law (M,X)(M ′, X ′) := (MM ′, XM ′ +X ′) and this group is called the full Jacobi

group.

Next I define actions of the groups SL2(Z) and Z2 on functions φ : H× C→ C, the

so-called slash operators.

Definition 3.2. Let φ : H× C→ C. Then

(i)(φ|k,m [ a bc d ]

)(τ, z) := (cτ + d)−ke−

2πimcz2

cτ+d φ(aτ+bcτ+d

, zcτ+d

),

(∀ [ a bc d ] ∈ Γ

)(ii) (φ|m[λ, µ])(τ, z) := e2πm(λ

2τ+2λz)φ(τ, z + λτ + µ), (∀ (λ, µ) ∈ Z2)

If M,M ′ ∈ Γ and X,X ′ ∈ Z2, then one easily verifies the relations

(φ|k,mM)|k,mM ′ = φ|k,m(MM ′),

(φ|mX)|mX ′ = φ|m(X +X ′),

and

(φ|k,mM)|mXM = (φ|mX)|k,mM.

These relations show that (i) and (ii) jointly define an action of the full Jacobi group.

Now I in a position to define Jacobi forms on the full Jacobi group.

8

Definition 3.3. A holomorphic function φ : H × C → C is a Jacobi form on Γ, of weight

k, and index m if the following conditions hold:

(i) φ|k,mM = φ ∀M ∈ Γ,

(ii) φ|mX = φ ∀X ∈ Z2,

and φ has a Fourier series expansion of the form

(iii)

φ(τ, z) =∞∑n=0

∑r∈Z

4nm−r2≥0

c(n, r)qnζr,

where here and throughout q := e2πiτ and ζ := e2πiz.

A Jacobi form is called a cusp form if its Fourier series expansion satisfies c(n, r) = 0 unless

4mn− r2 > 0. I denote the space of Jacobi forms of weight k and index m by Jk,m, and the

space of cusp form in Jk,m by Jcuspk,m .

The above definition is for the Jacobi forms for the full Jacobi group, but it can easily

be extended to subgroups (see also [7]).

Example 3.4. Let k ≥ 4. The Jacobi Eisenstein series of weight k and index m are defined

by

Ek,m(τ, z) :=∑

γ∈ΓJ∞\ΓJ1|k,m, where ΓJ∞ :=

{[± [ 1 n0 1 ] , (0, µ)

]|n, µ ∈ Z

}.

Then Ek,m ∈ Jk,m and one finds that

Ek,m(τ, z) =1

2

∑c,d∈Z

(c,d)=1

∑λ∈Z

(cτ + d)−ke2πim(λ2 aτ+b

cτ+d+2λ z

cτ+d− cz

2

cτ+d

).

Let en,r denote the Fourier series coefficients of Ek,m. In the special case that m = 1 one

can show that ek,1(n, r) =H(k−1,4n−r2)

ζ(3−2k) , where H(k− 1, N) is Cohen’s function [3]. Using the

values of Cohen’s function one obtains:

E4,1 = 1 + (ζ2 + 56ζ + 126 + 56ζ−1 + ζ−2)q + (126ζ2 + 576ζ + 756 + 576ζ−1 + 126ζ−2)q2 + ...

E6,1 = 1+(ζ2−88ζ−330−88ζ−1 +ζ−2)q+(−330ζ2−4224ζ−7524−4224ζ−1−330ζ−2)q2 + ...

9

One can use the Jacobi-Eisenstein series to define the two important cusp forms

φ10,1 =1

144(E6E4,1 − E4E6,1) ∈ J cusp10,1 ,

φ12,1 =1

144(E24E4,1 − E6E6,1) ∈ J

cusp12,1 ,

which have the following Fourier series expansions:

φ10,1 = (ζ − 2 + ζ−1)q + (−2ζ2 − 16ζ + 36− 16ζ−1 − 2ζ−2)q2 + ...

φ12,1 = (ζ + 10 + ζ−1)q + (10ζ2 − 88ζ − 132− 88ζ−1 + 10ζ−2)q2 + ...

Note that φ12,1φ10,1

is a meromorphic Jacobi form of weight 2 and index 0. In fact, one finds that

φ12,1(τ, z)

φ10,1(τ, z)=ζ + 10 + ζ−1

ζ − 2 + ζ−1+ 12(ζ − 2 + ζ−1)q + ...

is −3π2

times the Weierstrass ℘-function.

I end this section with a result on the Fourier series coefficients of Jacobi forms.

Theorem 3.5. Let φ be a Jacobi form of index m with Fourier development∑c(n, r)qnζr.

Then c(n, r) depends only on 4nm− r2 and on r (mod 2m). If k is even and m = 1 or if m

is prime, then c(n, r) depends only on 4nm− r2. If m = 1 and k is odd, then φ is identically

zero.

3.2. Taylor Development of Jacobi Forms and the Structure Theorem for Jk,1

In this section, I discuss Taylor coefficients of Jacobi forms, which are a tool in

determining the structure theorem of Jk,1. Consider the Taylor series of a Jacobi form

φ ∈ Jk,m around z = 0:

φ(τ, z) =∞∑ν=0

χν(τ)zν .

Then χ0 is a modular form of weight k, but χν is not modular if ν > 0. However, one can

use derivatives of the χν to construct modular forms. For example ζ2 := χ2 − 2πimk χ′0 is a

modular form of weight k + 2. This idea extends, and one can prove that

ζν(τ) :=∑

0≤µ≤ ν2

(−2πim)µ(k + ν − µ− 2)!(k + 2ν − 2)!µ!

χ(µ)ν−2µ(τ)

10

and

Dνφ(τ, z) :=(2πi)−ν(k + 2ν − 2)!(2ν)!

(k + ν − 2)!ζ2ν(τ)

are modular forms of weight k + ν on the full modular group Γ, and they are cusp forms if

ν > 0.

The next proposition follows from the elliptic transformation law (property (ii) of

Definition 3.3) of Jacobi forms and the argument principle in complex analysis.

Proposition 3.6. Let φ ∈ Jk,m. Then for fixed τ ∈ H, the function z 7→ φ(τ, z), if not

identically zero, has exactly 2m zeros (counting multiplicity) in any fundamental domain for

the action of the lattice Zτ + Z on C.

Hence Jacobi forms of index m are uniquely determined by their first 2m Taylor

coefficients. This fact and the definition of Dν imply the following theorems:

Theorem 3.7. Let φ ∈ Jk,m. Then the following map is injective:

D :=2m⊕ν=0

Dν →Mk(Γ)⊕ Sk+1(Γ)⊕ ...⊕ Sν(Γ).

Theorem 3.8. The following maps are isomorphisms:

Mk−4 ⊕Mk−6 → Jk,1,

(f, g) 7→ (fE4,1 + gE6,1)

D0 +D2 : Jk,1 →Mk ⊕ Sk+2,

φk,1 7→ D0φk,1 ⊕D2φk,1.

The final theorem in this section gives the structure of Jacobi cusp forms of index 1.

Theorem 3.9. The map

Mk−10 ⊕Mk−12 → J cuspk,1 ,

(f, g) 7→ (fφ10,1 + gφ12,1)

is an isomorphism, where φ10,1 and φ12,1 are the cusp forms introduced in Example 3.4.

11

3.3. The Theta Decomposition

Theorem 3.5 of Section 3.1 asserts that the coefficients c(n, r) of a Jacobi form of

index m depend only on the discriminant 4nm− r2 and on the value of r (mod 2m), i.e.,

c(n, r) = c(4nm − r2), cr′(N) = cr(N) for r′ ≡ r (mod 2m). This leads to the theta

decomposition of a Jacobi form.

Theorem 3.10. Let φ =∑

n,r c(n, r)qnζn ∈ Jk,m. Then

φ(τ, z) =∑

µ (mod 2m)

hµ(τ)θm,µ(τ, z),

where

hµ(τ) :=∞∑N=0

cµ(N)qN/4m with cµ(N) = c

(N + r24m

, r), (any r ∈ Z, r ≡ µ (mod 2m)),

cµ(N) = 0 if N 6≡ −µ2 (mod 4m),

and

θm,µ(τ, z) :=∑r∈Z

r≡µ (mod 2m)

qr2/4mζr.

I end this Chapter by pointing out that the theta decomposition links Jacobi forms

to half-integral weight modular forms (for details see §5 of [7]). In particular, if φ ∈ Jk,1,

then

φ(τ, z) =∑

µ (mod 2)

hµ(τ)θ1,µ(τ, z) = h0(τ)θ1,0(τ, z) + h1(τ)θ1,1(τ, z),

and it is easy to verify that h(τ) := h0(4τ) + h1(4τ) satisfies the following transformation

formulas:

(i) h(τ + 1) = h(τ).

(ii) h(

τ4τ+1

)= (4τ + 1)k−

12h(τ).

Thus, one finds that h is a modular form of weight k− 12

on the congruence subgroup Γ0(4).

Let Mk− 12(Γ0(4)) denote the vector space of such forms and let

M+k− 1

2

(Γ0(4)) :={h ∈Mk− 1

2(Γ0(4))

∣∣∣h = ∞∑N=0

(−1)k−1≡0,1 (mod 4)

c(N)qN}

12

be the Kohnen plus space. Then the precise link between Jacobi forms of index 1 and

half-integral weight modular forms is given by the following theorem.

Theorem 3.11. Let k be an even integer. Then

M+k− 1

2

(Γ0(4)) ∼= Jk,1(Γ)

where the isomorphism is given by∑N≥0

N≡0,3 (mod 4)

a(N)qN 7−→∑n,r∈Z

4n−r2≥0

a(4n− r2)qnζr.

13

CHAPTER 4

HERMITIAN JACOBI FORMS

Jacobi forms connect different types of automorphic forms, and in particular, they

appear as Fourier-Jacobi coefficients of Siegel modular forms of degree 2. Analogously,

Hermitian Jacobi forms appear as Fourier-Jacobi coefficients of Hermitian modular forms of

degree 2 over a complex quadratic field. In this chapter, I restrict myself to the case where

the complex quadratic field is the Gaussian number field Q(i).

Hermitian Jacobi forms over Q(i) were first introduced by Haverkamp [9, 10]. Later

Sasaki [25] and Das [4, 5] contributed further to the theory of such Jacobi forms. Unfor-

tunately, the existing notion of Hermitian Jacobi forms does not allow certain arithmetic

applications such as the study of so-called heat cyles. In this chapter, I extend the defi-

nition of Hermitian Jacobi forms, and I prove a structure theorem for this new space. As

an application, I will explore heat cycles and U(p)-conguences of Hermitian Jacobi forms in

Chapter 5. Throughout this chapter, k and m are again nonnegative integers, and if s ∈ C,

then s denotes its complex conjugate.

4.1. The Hermitian Jacobi Group and Hermitian Jacobi Forms

I first define the Hermitian Jacobi group. Then I introduce Hermitian Jacobi forms

of parity δ, and I define the space of Hermitian Jacobi forms as a direct sum of Hermitian

Jacobi forms of positive parity and Hermitian Jacobi forms of negative parity.

Let O := Z[i] be the ring of Gaussian integers, O× := {1,−1, i,−i} its group of units,

and Γ(O) :={�M | � ∈ O×, M ∈ SL2(Z)

}be the Hermitian modular group. Now I can

define the Hermitian Jacobi group.

Definition 4.1. The set ΓJ(O) := Γ(O) nO2 ={

(�M,X)|�M ∈ Γ(O), X ∈ O2}

forms a

group under the group law (�M,X)(�′M ′, X ′) := (��′MM ′, X(�′M ′) +X ′) and this group is

called the Hermitian Jacobi group.

I define the following slash operators on functions φ : H× C2 → C.

14

Definition 4.2. Let φ : H× C2 → C. Then

(i)

(φ|k,m,δ (�M)

)(τ, z, w) := σ(�)�−k(cτ + d)−ke−

2πimczwcτ+d φ

(Mτ,

�z

cτ + d,�−1w

cτ + d

)∀�M ∈ Γ(O)

(ii)

(φ|m[λ, µ]

)(τ, z, w) := e2πim(λλτ+zλ+λw)φ(τ, z + λτ + µ,w + λτ + µ) ∀[λ, µ] ∈ O2,

where, here and throughout δ = + if σ(�) = 1 and δ = − if σ(�) = �2.

If �M, �′M ′ ∈ Γ(O) and X,X ′ ∈ O2, then one can verify the relations

(φ|k,m,δ �M)|k,m,δ �′M ′ = φ|k,m,δ(�M�′M ′),

(φ|mX)|mX ′ = φ|m(X +X ′),

and

(φ|k,m,δ �M)|mX�M = (φ|mX)|k,m,δ �M.

These relations show that (i) and (ii) jointly define an action of the Hermitian Jacobi group.

I now in a position to define Hermitian Jacobi forms of parity δ.

Definition 4.3. A holomorphic function φδ : H × C2 → C is a Hermitian Jacobi form on

Γ(O), of weight k, index m, and of parity δ if the following conditions hold:

(i) φδ|k,m,δ(�M) = φ ∀ �M ∈ Γ(O),

(ii) φδ|m[λ, µ] = φ ∀ [λ, µ] ∈ O2,

(iii) and φδ has a Fourier series expansion of the form

φδ(τ, z, w) =∞∑n=0

∑r∈O#

nm−|r|2≥0

c(n, r)qnζr(ζ ′)r,

where as before q := e2πiτ and ζ := e2πiz, and also ζ′

:= e2πiw and O# := i2O is the

inverse different of Q(i).

15

A Hermitian Jacobi form is called a cusp form if its Fourier series coefficients vanish

unless mn − |r|2 > 0. I denote the space of Hermitian Jacobi forms of weight k, index m,

and of parity δ by Jδk,m(O), and the space of cusp form in Jδk,m(O) by Jδ,cuspk,m (O).

Remark 4.4. The space of Hermitian Jacobi forms in [9, 25, 4, 5] coincides with the space

of Hermitian Jacobi forms of positive parity, i.e., with J+k,m(O).

Next I define the space of Hermitian Jacobi forms as a direct sum of the spaces of

Hermitian Jacobi forms of positive and negative parity.

Definition 4.5. The space of Hermitian Jacobi forms of weight k and index m is defined

by

Jk,m(O) := J+k,m(O)⊕ J−k,m(O) =

{(φ+, φ−) | φ+ ∈ J+k,m(O), φ

− ∈ J−k,m(O)}.

The following proposition is an extension of Propositions 1.3 and 1.4 of [9].

Proposition 4.6. Let φδ be a Hermitian Jacobi form of weight k, index m, and parity δ

with Fourier series expansion∑c(n, r)qnζr(ζ ′)r. Then I have the following:

(i) The coefficient c(n, r) depends only on nm− |r|2 and on r (mod mO).

(ii) For any � ∈ O×, σ(�)�kc(n, r) = c(n, �r).

(iii) If m = 1, k ≡ 0 (mod 4) and δ = +, then c(n, r) depends only on n− |r|2.

(iv) If m = 1, k ≡ 2 (mod 4) and δ = −, then c(n, r) depends only on n− |r|2.

(v) If m = 1 and k is odd, then φδ is identically zero.

Proof. Assume that r ≡ r′ (mod mO), and nm− |r|2 = n′m− |r′|2.

If r′ = r +mλ (λ ∈ O), then

n′m− |r′|2 = n′m− |r|2 −m(λr + λr)−m2|λ|2 = nm− |r|2

implies that n′m = nm+mλr +mλr +m2|λ|2, i.e., n′ = n+ λr + λr +m|λ|2.

The elliptic transformation law (see (ii) of Definition 4.3) with λ = 0 yields

∑c(n, r)qnζr(ζ ′)r = qm|λ|

2

ζλm(ζ ′)λm∑

c(n, r)qn+λr+λrζr(ζ ′)re2πi(µr+µr)

16

=∑

c(n, r)qn+λr+λr+m|λ|2

ζλm+r(ζ ′)λm+r.

Thus, I find that c(n, r) = c(n+ λr + λr +m|λ|2, r + λm) = c(n′, r′), and (i) is proved.

The modular transformation law (see (i) of Definition 4.3) with �M = ( � 00 � ) asserts

that

φδ(τ, �z, �−1w) = σ(�)�kφδ(τ, z, w),

which implies that

∑c(n, r)qnζ�r(ζ ′)�r = σ(�)�k

∑c(n, r)qnζr(ζ ′)r.

Hence σ(�)�kc(n, r) = c(n, �r), and (ii) holds.

If m = 1 and n−|r2| = n′−|r′|2 then r ≡ r′ (mod O) or r ≡ ir′ (mod O). In the first

case, c(n, r) = c(n′, r′) by (i), and in the second case, c(n, r) = c(n′, ir′) = σ(−i)(−i)kc(n′, r′).

Therefore, if k ≡ 0 (mod 4) and δ = +, then c(n, r) = c(n′, r′) and if k ≡ 2 (mod 4)

and δ = −, then c(n, r) = c(n′, r′). I conclude that (iii) and (iv) hold.

Finally, if m = 1 and k is odd, then the modular transformation law with � = 1 and

M =( −1 0

0 −1)

guarantees that c(n, r) = −c(n,−r). Note that n − |r|2 = n − | − r|2 and

r ≡ −r (mod O), and hence c(n, r) = −c(n, r), i.e., c(n, r) = 0. �

4.2. The Theta Decomposition

In this section, I discuss the theta decomposition of Hermitian Jacobi forms. I first

introduce necessary notation. Consider a Hermitian Jacobi form φδ =∑c(n, r)qnζr(ζ ′)r.

Then for s ∈ O#/mO I define functions cs : Z→ C by

cs(N) =

c(N+4|r|2

4m, r)

if N ≡ −4|s|2 (mod 4m)

0 otherwise,

where r ∈ O#, r ≡ s (mod mO). Moreover, set

hs(τ) :=∞∑N=0

cs(N)qN/4m (r ∈ O#, r ≡ s (mod mO))(5)

17

and

θHm,s(τ, z, w) :=∑r∈O#

r≡s (mod mO)

q|r|2/mζr(ζ ′)r.

Exactly as in Haverkamp [9] one verifies that φδ =∑

n,r c(n, r)qnζr(ζ ′)r ∈ Jδk,m(O) has the

theta decomposition:

(6) φδ(τ, z, w) =∑

s∈O#/mO

hs(τ)θHm,s(τ, z, w).

Haverkamp [9] determines the transformation laws of the theta functions θHm,s. The

following proposition is Corollary 4.4 of [9].

Proposition 4.7. Set I = ( 1 00 1 ) , T = (1 10 1 ) , and J = (

0 1−1 0 ). Then

θHm,s∣∣m

[λ, µ] = θHm,s for λ, µ ∈ O,

θHm,s∣∣1,m,+

�I = �θHm,�s for � ∈ O×,

θHm,s∣∣1,m,+

T = e2πi|s|2m θHm,s,

θHm,s∣∣1,m,+

J =i

2m

∑t∈O#/mO

e4πiRe(st)

m θHm,t.

Again, as in Haverkamp [9] one finds that the functions hs in (6) are certain vector-

valued modular forms. I do not need detailed information about hs, and I only record the

following consequences of (6) and Proposition 4.7.

Corollary 4.8. Let φδ ∈ Jδk,m(O) and hs as in (6). Then

ikh−is(τ) =

hs(τ) if δ = +

−hs(τ) if δ = −

hs(τ + 1) = e− 2πi|s|

2

m hs(τ).

Proof. The modular transformation law (see (i) of Definition 4.3) with �M = ( i 00 i ) implies

that

(7) φδ(τ, iz,−iw) = σ(i)ikφδ(τ, z, w).

18

Note that the second transformation of Proposition 4.7 asserts that θHm,s(τ, iz,−iw) =

θHm,−is(τ, z, w), and comparing the theta decompositions in both sides of (7) yields the first

identity.

Similarly, the second identity follows from comparing the theta decompositons in both

sides of

φδ(τ + 1, z, w) = φδ(τ, z, w)

or alternatively, from using the congruence N ≡ −4|s|2 (mod 4m) in the definition of (5).

�

From here on I treat only Hermitian Jacobi forms of weight k and index 1. Consider

the set {0, 12, i

2, 1+i

2} of representatives for the set of cosets O#/O. The following lemma is

a generalization of Lemma 2 of [25], and it is an immediate consequence of the first relation

of Corollary 4.8.

Lemma 4.9. Let φδ ∈ Jδk,1(O) and hs as in (6).

If k ≡ 0 (mod 4) and δ = +, then h i2(τ) = h 1

2(τ).

If k ≡ 2 (mod 4) and δ = +, then h0(τ) = h 1+i2

(τ) = 0, h i2(τ) = −h 1

2(τ).

If k ≡ 0 (mod 4) and δ = −, then h0(τ) = h i+12

(τ) = 0, h i2(τ) = −h 1

2(τ).

If k ≡ 2 (mod 4) and δ = −, then h i2(τ) = h 1

2(τ).

I now recall the theta decompositions for specific examples of Hermitian Jacobi forms

of positive parity. Let x, y, z be the theta constants that I introduced in (3) of Chapter 2.

The Hermitian Jacobi forms φ+k,1 ∈ J+k,1(O) for k = 4, 8, 12, 16 and φ

+, cusp10,1 ∈ J

+, cusp10,1 (O)

19

were considered in [25] and it was shown that

φ+4,1 =1

2(x6 + y6)θH1,0 +

1

2z6(θH

1, 12

+ θH1, i

2) +

1

2(x6 − y6)θH

1, 1+i2

φ+8,1 =1

2(x14 + y14)θH1,0 +

1

2z14(θH

1, 12

+ θH1, i

2) +

1

2(x14 − y14)θH

1, 1+i2

φ+12,1 =1

2(x22 + y22)θH1,0 +

1

2z22(θH

1, 12

+ θH1, i

2) +

1

2(x22 − y22)θH

1, i+12

φ+16,1 =1

2(x30 + y30)θH1,0 +

1

2z30(θH

1, 12

+ θH1, i

2) +

1

2(x30 − y30)θH

1, 1+i2

φ+, cusp10,1 =1

64

(x6y6z6θH

1, 12− x6y6z6θH

1, i2

).

(8)

In Remark 4.11 I will also give the initial Fourier series expansions of these Hermitian

Jacobi forms.

Note that [25] uses the Hermitian Jacobi forms φ+4,1, φ+8,1, φ

+12,1 and φ

+, cusp10,1 to de-

termine the structure of J+k,1(O) and the cusp forms ψ+, cuspk,1 ∈ J

+, cuspk,1 (O) for k = 8, 12, 16

to determine the structure of J+, cuspk,1 (O). The cusp forms ψ+, cuspk,1 ∈ J

+, cuspk,1 (O) for k =

8, 12, 16 are defined by

ψ+, cusp8,1 := E4φ+4,1 − φ+8,1

ψ+, cusp12,1 := E4φ+8,1 − φ+12,1

ψ+, cusp16,1 := E4φ+12,1 − φ+16,1,

where E4 is again the usual modular Eisenstein series of weight 4.

Note that the above examples of Hermitian Jacobi forms differ from the definitons

in [25] by some multiplicative scalars.

Hermitian Jacobi forms of negative parity have not been studied rigorously in the

literature, but they do arise via Fourier-Jacobi coefficients of Hermitan modular forms of

degree 2 with certain characters (see [6]). Hermitian Eisenstein series are examples of such

Hermitian modular forms of degree 2. In particular, there exists such a Hermitian Eisenstein

series of weight 6, whose first Fourier-Jacobi coefficient φ−6,1 is a Hermitian Jacobi form of

negative parity, weight 6, and index 1. It is somewhat difficult to explicitly compute the

Fourier series coefficients of Hemitian Eisenstein series. I determined φ−6,1 using a different

20

approach. I used SAGE [28] and SAGE code written by Martin Raum to calculate several

Fourier series coefficients of φ−6,1. This allowed us to guess the theta decomposition of φ−6,1,

which I then verified directly. The Hermitian Jacobi form φ−6,1 will play an important role in

the next section.

Lemma 4.10. Let x, y, and z be again the classical theta constants. Then

φ−6,1 := h0θH1,0 + h 1

2θH

1, 12

+ h i2θH

1, i2

+ h 1+i2θH

1, 1+i2∈ J−6,1(O),

where

h0 := −1

2(x2 + y2)(x8 − x6y2 − x4y4 − x2y6 + y8),

h 12

:=1

2z6(z4 − 2x4),

h i2

:=1

2z6(z4 − 2x4),

h 1+i2

:= −12

(x2 − y2)(x8 + x6y2 − x4y4 + x2y6 + y8).

Proof. Consider ψ+12,1 :=152E4φ

+8,1 − 2E24φ+4,1 − 92φ

+12,1. Then ψ

+12,1 ∈ J+12,1(O), and let

ĥ0θH1,0 + ĥ 1

2θH

1, 12

+ ĥ i2θH

1, i2

+ ĥ 1+i2θH

1, 1+i2

be the theta decomposition of ψ+12,1. Recall the following

identities from Chapter 2:

x4 = y4 + z4, E4 =12(x8 + y8 + z8), E6 =

12(x4 + y4)(x4 + z4)(y4 − z4).

These identities allow us (with the help of Mathematica) to verify that

ĥ0 = E6h0

ĥ 12

= E6h 12

ĥ i2

= E6h i2

ĥ 1+i2

= E6h 1+i2

.

Hence ψ+12,1 = E6φ−6,1. Observe that the modular Eisenstein series E6 can also be viewed

as a weight 6 and index 0 Hermitian Jacobi form of negative parity. I conclude that φ−6,1 ∈

J−6,1(O). �

I end this section with the initial Fourier series expansions of the Hermitian Jacobi

forms φ+4,1, φ−6,1, φ

+8,1, and φ

+,cusp10,1 . See also the Appendix for more coefficients of these forms.

21

Remark 4.11. I have the following initial Fourier series expansions:

φ+4,1 = 1 + q(

60 + 32(ζ

12 (ζ ′)

12 + ζ−

12 (ζ ′)−

12 + ζ−

i2 (ζ ′)

i2 + ζ

i2 (ζ ′)−

i2

)+(ζζ ′ + ζ−1(ζ ′)−1 + ζ−i(ζ ′)i + ζ i(ζ ′)−i

)+ 12

(ζ

1+i2 (ζ ′)

1−i2 + ζ

−1+i2 (ζ ′)

−1−i2 + ζ

1−i2 (ζ ′)

1+i2 + ζ

−1−i2 (ζ ′)

−1+i2

))+ · · ·

φ−6,1 = 1 + q(− 204− 64

(ζ

12 (ζ ′)

12 + ζ−

12 (ζ ′)−

12 + ζ−

i2 (ζ ′)

i2 + ζ

i2 (ζ ′)−

i2

)+(ζζ ′ + ζ−1(ζ ′)−1 + ζ−i(ζ ′)i + ζ i(ζ ′)−i

)− 12

(ζ

1+i2 (ζ ′)

1−i2 + ζ

−1+i2 (ζ ′)

−1−i2 + ζ

1−i2 (ζ ′)

1+i2 + ζ

−1−i2 (ζ ′)

−1+i2

))+ · · ·

φ+8,1 = 1 + q(

364 +(ζζ ′ + ζ−1(ζ ′)−1 + ζ−i(ζ ′)i + ζ i(ζ ′)−i

)+ 28

(ζ

1+i2 (ζ ′)

1−i2 + ζ

−1+i2 (ζ ′)

−1−i2 + ζ

1−i2 (ζ ′)

1+i2 + ζ

−1−i2 (ζ ′)

−1+i2

))+ · · ·

φ+,cusp10,1 = q(ζ

12 (ζ ′)

12 + ζ−

12 (ζ ′)−

12 − ζ−

i2 (ζ ′)

i2 − ζ

i2 (ζ ′)−

i2

)+q2

(− 18

(ζ

12 (ζ ′)

12 + ζ−

12 (ζ ′)−

12 − ζ−

i2 (ζ ′)

i2 − ζ

i2 (ζ ′)−

i2

)+(ζ

1+2i2 (ζ ′)

1−2i2 + ζ

−1+2i2 (ζ ′)

−1−2i2 + ζ

1−2i2 (ζ ′)

1+2i2 + ζ

−1−2i2 (ζ ′)

−1+2i2

− ζ2+i2 (ζ ′)

2−i2 − ζ

−2+i2 (ζ ′)

−2−i2 − ζ

2−i2 (ζ ′)

2+i2 − ζ

−2−i2 (ζ ′)

−2+i2

))+ · · ·

4.3. The Structure of Hermitian Jacobi Forms of Index 1

In this section, I determine the structure of Hermitian Jacobi forms of index 1. I

find that the ring of Hermitian Jacobi forms of index 1 is a free module of rank 4 over the

ring of modular forms, and a set of generators is given by the forms φ+4,1, φ−6,1, φ

+8,1, φ

+,cusp10,1 .

In particular, if I restrict to forms of positive parity, then I recover the structure result of

Sasaki [25] as a special case. My approach is based on the methods in [7] and [25], and I

begin by investigating the Tayor series coefficients of Hermitian Jacobi forms.

22

Consider the Taylor series expansion of a Hermitian Jacobi form φδ ∈ Jδk,1(O) around

(z, w) = (0, 0):

φδ(τ, z, w) =∞∑

µ,ν=0

χδµ,ν(τ)zµwν .

The modular transformation law (see (i) of Definition 4.3) with �M = � ( a bc d ) implies that

χδµ,ν

(aτ + bcτ + d

)= σ(ε)εk−µ+ν(cτ + d)k+µ+ν

(χδµ,ν(τ) +

2πic

cτ + dχδµ−1,ν−1(τ) +

1

2!

( 2πiccτ + d

)2χδµ−2,ν−2(τ) + ...

).

Observe that χδµ,ν = 0 unless µ − ν is even. The first several coefficients have the following

behavior under modular transformations:

χδ0,0

(aτ + bcτ + d

)= σ(ε)εk(cτ + d)kχδ0,0(τ),

χδ1,1

(aτ + bcτ + d

)= σ(ε)εk

{(cτ + d)k+2χδ1,1(τ) + 2πic(cτ + d)

k+1χδ0,0(τ)},

χδ0,2

(aτ + bcτ + d

)= σ(ε)εk+2(cτ + d)k+2χδ0,2(τ),

χδ2,0

(aτ + bcτ + d

)= σ(ε)εk+2(cτ + d)k+2χδ2,0(τ),

χδ0,4

(aτ + bcτ + d

)= σ(ε)εk+4(cτ + d)k+4χδ0,4(τ),

χδ2,2

(aτ + bcτ + d

)= σ(ε)εk

((cτ + d)k+4χδ2,2(τ) + 2πic(cτ + d)

k+3χδ1,1(τ)

+1

2!(2πic)2(cτ + d)k+2χδ0,0(τ)

).

The following proposition is an immidiate consequence of the above equations.

Proposition 4.12. Let φδ(τ, z, w) =∑∞

µ,ν=0 χδµ,ν(τ)z

µwν ∈ Jδk,1(O).

If k ≡ 0 (mod 4) and δ = +, then χ+0,2(τ) = χ+2,0(τ) = 0.

If k ≡ 0 (mod 4) and δ = −, then χ−0,0(τ) = χ−1,1(τ) = χ−4,0(τ) = χ−0,4(τ) = χ−2,2(τ) = 0.

If k ≡ 2 (mod 4) and δ = −, then χ−0,2(τ) = χ−2,0(τ) = 0.

If k ≡ 2 (mod 4) and δ = +, then χ+0,0(τ) = χ+1,1(τ) = χ+4,0(τ) = χ+0,4(τ) = χ+2,2(τ) = 0.

23

Observe that the Fourier series expansion of a Hermitian Jacobi form

φδ(τ, z, w) =∞∑n=0

∑r∈O#

nm−|r|2≥0

c(n, r)qnζr(ζ ′)r ∈ Jδk,1(O)

implies that its Taylor series coefficients χδµ,ν have a q-expansion of a holomorphic modular

form:

χδµ,ν(τ) =(πi)µ+ν

µ!ν!

( ∞∑r∈O#

n−|r|2≥0

rµrνc(n, r))qn.

I state the following proposition, where Mk and Sk denote again the weight k spaces of

modular forms and cusp forms, respectively.

Proposition 4.13. I have

χδ(0,0)(τ) ∈Mk,

χδ(2,0)(τ), χδ(0,2)(τ) ∈ Sk+2,

χδ(4,0)(τ), χδ(0,4)(τ) ∈ Sk+4,

ζδ1.1(τ) := χδ1,1(τ)− 2πik (χ

δ(0,0))

′(τ) ∈ Sk+2,

ζδ2.2(τ) := χδ2,2(τ)− 2πik+2(χ

δ(1,1))

′(τ) + (2πi)2

2(k+1)(k+2)(χδ(0,0))

′′(τ) ∈ Sk+4.

Recall the definition of the Jacobi theta function in (2) of Chapter 2:

θa,b(τ, z) =∑n∈Z

eπi(a+n)2τ+2πi(n+a)(z+b).

Note that θa,b(2τ, z + w) satisfies the heat equation, i.e.,

∂2

∂z2θa,b(2τ, z + w) = 2πi

∂

∂τθa,b(2τ, z + w).

Moreover, it is easy to see that θa,0(τ, z) is an even function of z. Hence θa,0(2τ, z + w) has

a Taylor series expansion of the form

θa,0(2τ) +2πi

2!

d

dτθa,0(2τ)(z + w)

2 +(2πi)2

4!

d2

dτ 2θa,0(2τ)(z + w)

4 + ...

and θb,0(2τ, i(w − z)) has a Taylor series expansion of the form

θb,0(2τ)−2πi

2!

d

dτθb,0(2τ)(w − z)2 +

(2πi)2

4!

d2

dτ 2θb,0(2τ)(w − z)4 + ...

24

For convenience I write

T2a := θa,0(2τ), T2a′ := 2πi

d

dτθa,0(2τ), T2a

′′ := (2πi)2d2

dτ 2θa,0(2τ),

and T2b, T2b′, and T2b

′′ are defined analogously. Then I have

θa,0(2τ, z + w)θb,0(2τ, i(w − z)) = T2aT2b + (T2aT2b′ + T2a′T2b)zw

+1

2(T2a

′T2b − T2aT2b′)(z2 + w2)

+1

4(T2a

′′T2b + T2aT2b′′ + 2T2a

′T2b′)z2w2

+1

4!(T2a

′′T2b + T2aT2b′′ − 6T2a′T2b′)(z4 + w4) + · · ·

Furthermore, one easily verifies the following factorization of theta functions:

(9) θa2,0(2τ, z + w)θ b

2,0(2τ, i(w − z)) = θ

H1,a+bi

2

(τ, z, w).

Now I expand the theta decomposition

φδ(τ, z, w) =∑a,b=0,1

ha+bi2

(τ)θH1,a+bi

2

(τ, z, w)

=(9)

∑a,b=0,1

ha+bi2

(τ)θa2,0(2τ, z + w)θ b

2,0(2τ, i(w − z))

into a Taylor series, and compare its coefficients with the coefficients of the Taylor series

φδ(τ, z, w) = χδ0,0(τ) + χδ1,1(τ)zw +

(χδ0,2(τ) + χ

δ2,0(τ)

)(z2 + w2)

+ χδ2,2(τ)z2w2 +

(χδ0,4(τ) + χ

δ4,0(τ)

)(z4 + w4) + · · · .

I obtain

(χδ0,0, χδ1,1, χ

δ2,0 + χ

δ0,2, χ

δ2,2, χ

δ4,0 + χ

δ0,4) = (h0, h 1

2, h i

2, h 1+i

2)A′,

where A′ equals the matrix T02 2T0T0

′ 0 12 (T0T0′′+T0

′2) 14! (2T0′′T0−6T0′

2)

T0T1 T0T1′+T0

′T112 (T1

′T0−T1T0′) 14 (T1T0′′+T0T1

′′+2T1′T0

′) 14! (T1T0′′+T0T1

′′−6T1′T0′)T0T1 T0T1

′+T0′T1 − 12 (T1

′T0−T1T0′) 14 (T1T0′′+T0T1

′′+2T1′T0

′) 14! (T1T0′′+T0T1

′′−6T1′T0′)T1

2 2T1T1′ 0 12 (T1

′′T1+T1′2) 14! (2T1

′′T1−6T1′2)

.Equivalently, I find that

(χδ0,0, χδ1,1, χ

δ2,0,

1

2(χδ2,2 − 12χδ4,0)) = (h0, h 1

2, h i

2, h 1+i

2)A,(10)

25

where

A :=

T0

2 2T0T0′ 0 T0

′2

T0T1 T0T1′ + T0

′T114(T1′T0 − T1T0′) 2T0′T1′

T0T1 T0T1′ + T0

′T1 −14(T1′T0 − T1T0′) 2T0′T1′

T12 2T1T1

′ 0 T1′2

.(11)

Note that

detA = −12

(T1T0′ − T0T1′)2((T1T0′)2 − 4T0T0′T1T1′).

Consider the q-expansions

T0 = 1 + 2q + 2q4 + · · ·

T ′0 = −8π2q − 32π2q4 − · · ·

T1 = 2q14 + 2q

94 + · · ·

T ′1 = −2π2q14 − 18π2q

94 − · · ·

to find that

detA 6= 0.

Now I am in a position to determine the structure of Hermitian Jacobi forms of index

1. As before, χδµ,ν denote the Taylor coefficients of a Hermitian Jacobi form, and ζδµ,ν are the

linear combinations of Taylor coefficients as in Proposition 4.13:

Theorem 4.14. Assume that k ≡ 0 (mod 4). Then both linear maps

ζ : Jk,1(O) = J+k,1(O)⊕ J−k,1(O)→Mk ⊕ Sk+2 ⊕ Sk+2 ⊕ Sk+4

φ 7→ (χ+0,0, ζ+1,1, χ−2,0, ζ+2,2 − 12χ+4,0)

and

η : Mk−4 ⊕Mk−6 ⊕Mk−8 ⊕Mk−10 → Jk,1(O) = J+k,1(O)⊕ J−k,1(O)

(e, f, g, h) 7→ (eφ+4,1 + fφ−6,1 + gφ+8,1, hφ+,cusp10,1 )

are isomorphisms.

26

Assume that k ≡ 2 (mod 4). Then both linear maps

ζ : Jk,1(O) = J+k,1(O)⊕ J−k,1(O)→Mk ⊕ Sk+2 ⊕ Sk+2 ⊕ Sk+4

φ 7→ (χ−0,0, ζ−1,1, χ+2,0, ζ−2,2 − 12χ−4,0)

and

η : Mk−4 ⊕Mk−6 ⊕Mk−8 ⊕Mk−10 → Jk,1(O) = J+k,1(O)⊕ J−k,1(O)

(e, f, g, h) 7→ (hφ+,cusp10,1 , eφ+4,1 + fφ−6,1 + gφ+8,1)

are isomorphisms.

Proof. I only prove the first case, and the proof of the case k ≡ 2 (mod 4) is completely

analogous. Note that Proposition 4.13 shows that the map ζ is well-defined. First, I show

the injectivity of ζ. Let φ = (φ+, φ−) ∈ Jk,1(O). If ζ(φ) = 0, then χ+0,0 = ζ+1,1 = χ−2,0 =

ζ+2,2 − 12χ+4,0 = 0. Then Proposition 4.13 implies that

χ+1,1 = 0, ζ+2,2 = χ

+2,2, and hence

1

2(χ+2,2 − 12χ+4,0) = 0 .

Moreover, Proposition 4.12 gives that χ+2,0 = 0 and χ−0,0 = χ

−1,1 = χ

−2,2 = χ

−4,0 = 0.

Thus, for δ = ± I find that

(0, 0, 0, 0) = (χδ0,0, χδ1,1, χ

δ2,0,

1

2(χδ2,2 − 12χδ4,0)) =

(10):(h0, h 1

2, h i

2, h 1+i

2)A,

where A as in (11). Recall that detA 6= 0. Hence φ = (0, 0), which proves the injectivity of

ζ.

Next I show the injectivity of η. Let (e, f, g, h) ∈Mk−4 ⊕Mk−6 ⊕Mk−8 ⊕Mk−10 and

suppose that eφ+4,1 + fφ−6,1 + gφ

+8,1 + hφ

+,cusp10,1 = 0. Observe the theta decompositions of φ

+4,1,

φ+8,1, and φ+,cusp10,1 in (8) and of φ

−6,1 in Lemma 4.10 to find that

(e, f, g, h)H = (0, 0, 0, 0),

27

where

H :=

12(x6 + y6) 1

2z6 1

2z6 1

2(x6 − y6)

h0 h 12

h i2

h 1+i2

12(x14 + y14) 1

2z14 1

2z14 1

2(x14 − y14)

0 164x6y6z6 − 1

64x6y6z6 0

with h0, h 1

2, h i

2, and h 1+i

2as in Lemma 4.10. With the help of Mathematica (observing the

identity x4 = y4 + z4) one finds that

detH = − 9128

x16y16z16.

Recall the q-expansions of the theta constants (3). Specifically,

x = 1 + 2q12 + · · ·

y = 1− 2q12 + · · ·

z = 2q18 + 2q

98 + · · ·

and I find that x16y16z16 6= 0 . Hence detH 6= 0, which shows e = f = g = h = 0.

Finally, Theorem 2.4 implies that

dimMk + dimSk+2 + dimSk+2 + dimSk+4 = dimMk−4 + dimMk−6 + dimMk−8 + dimMk−10,

and I conclude that ζ and η are isomorphisms. �

Remark 4.15. If I restrict the maps η and ζ in Theorem 4.14 to the case of positive parity,

then I recover the structure of J+k,1(O) given in Sasaki [25].

28

CHAPTER 5

U(p) CONGRUENCES OF HERMITIAN JACOBI FORMS

In Section 2.2 I briefly reviewed U(p) congruences of modular forms. I pointed out

that Tate’s theory of theta cycles (see §7 of [12]) implies Theorem 2.7, which provides a

criterion for the existence of such congruences. Richter [23, 24] has established similar results

for Jacobi forms, which Raum and Richter [22] have extended to the case of Jacobi forms

of higher degree. In this chapter, I proceed as in [22, 23, 24] to explore U(p) congruences

of Hermitian Jacobi forms of index 1, and I determine a criterion for the existence of such

congruences. Throughout, p ≥ 5 is a prime, k and m are nonnegative integers, and as in

Chapter 4 I consider Hermitian Jacobi forms associated to the Gaussian number field Q(i).

5.1. Congruences and Filtrations

In this section, I investigate congruences and filtrations of Hermitian Jacobi forms.

More specifically, I extend Tate’s theory of theta cycles to Hermitian Jacobi forms, which

yields a criterion for the existence of U(p) congruences of Hermitian Jacobi forms.

Consider the heat operator (see also [13])

(12) Lm := −1

π2

(2πim

∂

∂τ− ∂

2

∂w∂z

).

The following lemma gives its action on Hermitian Jacobi forms.

Lemma 5.1. If φ ∈ Jδk,m(O), then

Lm(φ) =(k − 1)m

3E2φ+ φ̂

where E2 is the quasimodular Eisenstein series, and where φ̂ ∈ J−δk+2,m(O).

Remark 5.2. Observe that the “corrected” heat operator L̃m := Lm − (k−1)m3 E2 sends

Hermitian Jacobi forms of parity δ to forms of parity −δ. This is the reason that I introduced

the new space of Hermitian Jacobi forms in Chapter 4.

29

Proof of Lemma 5.1. First I show that if φ = φ(τ, z, w) satisfies the modular property (i)

of Definition 4.3 with parity δ, then φ̂ := Lm(φ)− (k−1)m3 E2φ satisfies the modular property

with parity −δ. Let ε ( a bc d ) ∈ Γ(O) and set η :=(aτ+bcτ+d

, �zcτ+d

, �−1wcτ+d

). I have

∂2φ

∂w∂z(η) = (cτ + d)2

∂2

∂w∂zφ(η) = (cτ + d)2

∂2

∂w∂zσ(�)�k(cτ + d)ke

2πimczwcτ+d φ

= σ(�)�k(cτ + d)k+2∂

∂w

(∂φ∂z

+2πimcw

cτ + dφ)e

2πimczwcτ+d

= σ(�)�k(cτ + d)k+2e2πimczwcτ+d

(∂2φ

∂w∂z+

2πimc

cτ + d

(φ+ w

∂φ

∂w+ z

∂φ

∂z

)− 4π

2m2c2zw

(cτ + d)2φ

).

Furthermore,

∂

∂τφ(η) = σ(�)�ke

2πimczwcτ+d

((cτ + d)k−1kcφ− (cτ + d)k 2πimc

2zw

(cτ + d)2φ+ (cτ + d)k

∂φ

∂τ

)∂φ

∂z(η) = σ(�)�k−1(cτ + d)k+1e

2πicmzwcτ+d

(∂φ∂z

+2πimcw

cτ + dφ)

∂φ

∂w(η) = σ(�)�k+1(cτ + d)k+1e

2πimczwcτ+d

( ∂φ∂w

+2πimcz

cτ + dφ)(13)

and

(14)∂φ

∂τ(η) = (cτ + d)2

∂

∂τφ(η) + �cz

∂φ

∂z(η) + �−1cw

∂φ

∂w(η).

Substituting (13) in (14) yields

∂φ

∂τ(η) = σ(�)�k(cτ + d)ke

2πimczwcτ+d

((cτ + d)kcφ+ 2πimc2zwφ+ (cτ + d)2

∂φ

∂τ

+ (cτ + d)(cz∂φ

∂z+ cw

∂φ

∂w

)).

I find that

φ̂(η) = Lm(φ(η)

)− (k − 1)m

3E2

(aτ + bcτ + d

)φ(η)

=−2miπ

∂φ

∂τ(η) +

1

π2∂2φ

∂w∂z(η)− (k − 1)m

3E2

(aτ + bcτ + d

)φ(η).

Substituting ∂φ∂τ

(η), ∂2φ

∂w∂z(η), and using (1) of Chapter 2 gives

φ̂(η) = σ(�)�−2�k+2(cτ + d)k+2e2πimczwcτ+d φ̂,

i.e., φ̂ satisfies the modular property (i) of Definition 4.3 with weight k + 2 and parity −δ.

30

Next I show that if φ = φ(τ, z, w) satisfies the modular property (ii) of Definition 4.3,

then so does φ̂. Let [λ, µ] ∈ O2 and set γ := (τ, z+λτ+µ, ω+λτ+µ) and Λ := (λλτ+λz+λw).

By assumption,

(15) φ(γ) = e−2πimΛφ.

I have

∂

∂τφ(γ) =

∂φ

∂τ(γ)

∂τ

∂τ+∂φ

∂z(γ)

∂

∂τ(z + λτ + µ) +

∂φ

∂w(γ)

∂

∂τ(w + λτ + µ)

=∂φ

∂τ(γ) + λ

∂φ

∂z(γ) + λ

∂φ

∂w(γ),

i.e.,

(16)∂φ

∂τ(γ) =

∂

∂τφ(γ)− λ∂φ

∂z(γ)− λ ∂φ

∂w(γ).

On the other hand, by (15)

∂

∂τφ(γ) =

(∂φ∂τ− 2πimλλφ

)e−2πimΛ

∂

∂zφ(γ) =

(∂φ∂z− 2πimλφ

)e−2πimΛ

∂

∂wφ(γ) =

( ∂φ∂w− 2πimλφ

)e−2πimΛ.

(17)

Furthermore,

∂φ

∂z(γ) =

∂

∂zφ(γ)

∂φ

∂w(γ) =

∂

∂wφ(γ)

∂2φ

∂w∂z(γ) =

∂2

∂w∂zφ(γ).

(18)

Equations (16), (17), and (18) imply that

∂φ

∂τ(γ) =

(∂φ∂τ

+ 2πimλλφ− λ∂φ∂z− λ ∂φ

∂w

)e−2πimΛ

∂2φ

∂w∂z(γ) =

( ∂2φ∂w∂z

− 2πimλ ∂φ∂w− 2πimλ∂φ

∂z− 4π2m2λλφ

)e−2πimΛ,

(19)

31

and I find that

Lm(φ(γ)

)=−1π2

(2πim

∂φ

∂τ(γ)− ∂

2φ

∂w∂z(γ))

= e−2πimΛ(−1π2

)(2πim

∂φ

∂τ− ∂

2φ

∂w∂z

)= e−2πimΛLm(φ).

Hence

φ̂(γ) = Lm(φ(γ)

)− (k − 1)m

3E2(τ)φ(γ) = e

−2πimΛφ̂,

i.e., φ̂ satisfies the elliptic property. I conclude that φ̂ is a Hermitian Jacobi form of weight

k + 2, index m, and parity −δ. �

I now give the action of the heat operator on the four generators.

Example 5.3. Lemma 5.1 in combination with Remark 4.11 gives the following identities:

L(φ+4,1) = E2φ+4,1 − φ−6,1

L(φ−6,1) =5

3E2φ

−6,1 −

8

3E4φ

+4,1 + φ

+8,1

L(φ+8,1) =7

3E2φ

+8,1 −

14

9E6φ

+4,1 −

7

9E4φ

−6,1

L(φ+,cusp10,1 ) = 3E2φ+,cusp10,1

(20)

In the following, let m = 1, and for convenience, I write L := L1. I denote with Fp the

field Z/pZ, with Z(p) the ring of p-integral rationals, and with Jδk,1(Z(p)) the ring of Hermitian

Jacobi forms of weight k, index 1, parity δ, and with p-integral rational coefficients. For

Hermitian Jacobi forms φ(τ, z, w) =∑c(n, r)qnζr(ζ ′)r and ψ(τ, z, w) =

∑c′(n, r)qnζr(ζ ′)r

with p-integral rational coefficients, I write φ ≡ ψ (mod p) whenever c(n, r) ≡ c′(n, r)

(mod p) for all n, r.

Lemma 5.4. The generators φ+4,1, φ−6,1, φ

+8,1, and φ

+,cusp10,1 are linearly independent over Fp.

Proof. Let a, b, c, d ∈ Fp and assume that aφ+4,1+bφ−6,1+cφ+8,1+dφ+,cusp10,1 ≡ 0 (mod p). Recall

that Remark 4.11 gives the initial Fourier series expansions of φ+4,1, φ−6,1, φ

+8,1, and φ

+,cusp10,1 .

In particular, their coefficients of q0ζ0(ζ ′)0 are 1, 1, 1, 0, respectively, their coefficients of

q1ζ12 (ζ ′)

12 are 32, −64, 0, 1, respectively, their coefficients of q1ζ i2 (ζ ′)−i2 are 32, −64, 0, −1,

32

respectively, and their coefficients of q1ζ1−i2 (ζ ′)

1+i2 are 12, −12, 28, 0, respectively. Comparing

the coefficients of q1ζ12 (ζ ′)

12 and q1ζ

i2 (ζ ′)

−i2 gives the system of equations:

32a− 64b+ d ≡ 0 (mod p)

32a− 64b− d ≡ 0 (mod p),

and I find that d ≡ 0 (mod p). Comparing the coefficients of q0ζ0(ζ ′)0, q1ζ 1−i2 (ζ ′) 1+i2 , and

q1ζ12 (ζ ′)

12 leads to the following system of equations:

a+ b+ c ≡ 0 (mod p)

12a− 12b+ 28c ≡ 0 (mod p)

32a− 64b+ 0c ≡ 0 (mod p) .

Hence

16a+ 40b ≡ 0 (mod p)

32a− 64b ≡ 0 (mod p)

and since p 6= 2, 3, I find that a ≡ b ≡ c ≡ 0 (mod p), i.e., φ+4,1, φ−6,1, φ+8,1 , and φ+,cusp10,1 are

linearly independent over Fp. �

Proposition 5.5. If φ ∈ Jδk,1(Z(p)) such that φ = eφ+4,1 + fφ−6,1 + gφ+8,1 (or φ = hφ+,cusp10,1 ),

then the elliptic modular forms e, f , and g (or h) have p-integral rational coefficients.

Moreover, if φ ≡ 0 (mod p), then e ≡ f ≡ g ≡ 0 (mod p) (or h ≡ 0 (mod p)).

Proof. Suppose that φ = eφ+4,1 + fφ−6,1 + gφ

+8,1 (the case φ = hφ

+,cusp10,1 is analogous). Note

that the elliptic modular forms e, f , and g have bounded denominators. If e, f , or g do

not have p-integral rational coefficients, then there exists some integer t ≥ 1 such that

0 ≡ ptφ ≡ pteφ+4,1 + ptfφ−6,1 + ptgφ+8,1 (mod p). This yields a nontrivial linear dependence

relation for φ+4,1, φ−6,1, and φ

+8,1, which contradicts Lemma 5.4.

Similarly, if φ ≡ 0 (mod p) such that e, f , or g do not vanish modulo p, then one also

obtains a nontrivial linear dependence relation for φ+4,1, φ−6,1, and φ

+8,1, which again contradicts

Lemma 5.4.

33

�

An argument as in Lemma 2.1 of Sofer [27] shows that if Hermitian Jacobi forms of

indices m and m′ are congruent modulo p, then m = m′. The following corollary is analogous

to Sofer’s Lemma 2.1 in the case m = 1.

Corollary 5.6. Let φ ∈ Jδk,1(Z(p)) and ψ ∈ Jδ′

k′,1(Z(p)) such that 0 6≡ φ ≡ ψ (mod p). Then

k ≡ k′ (mod (p− 1)).

Proof. Recall that if two modular forms fi ∈Mki (i = 1, 2) have p-integral rational coeffi-

cients such that 0 6≡ f1 ≡ f2 (mod p), then k1 ≡ k2 (mod (p − 1)) (see [26, 29]). This fact

in combination with Proposition 5.5 implies the claim. �

Remark 5.7. Let φ ∈ Jδk,1(Z(p)) and ψ ∈ Jδ′

k′,1(Z(p)) such that φ ≡ ψ (mod p). If δ 6= δ′ and

k ≡ k′ (mod 4), then φ ≡ ψ ≡ 0 (mod p).

Corollary 5.6 shows that there are congruences among Hermitian Jacobi forms of

different weights. Hence it is desirable to find the smallest weight in which the (coefficient-

wise) reduction of a Hermitian Jacobi form modulo p exists.

Definition 5.8. Set J̃δk,1 :={φ (mod p) : φ ∈ Jδk,1(Z(p))

}. For Hermitian Jacobi forms

with p-integral rational coefficients, I define the filtration modulo p by

Ω(φ) := inf{k : φ (mod p) ∈ J̃δk,1

}.

Next I define the U(p) operator for Hermitian Jacobi forms.

Definition 5.9. For φ(τ, z, w) =∑

n∈Z,r∈O#nm−|r|2≥0

c(n, r)qnζr(ζ ′)r, I define:

φ(τ, z, w)∣∣∣U(p) := ∑

n∈Z,r∈O#nm−|r|2≥0p|4(nm−|r|2)

c(n, r)qnζr(ζ ′)r.

34

Observe that if φ =∑c(n, r)qnζr(ζ ′)r, then Lm(φ) =

∑4(nm− |r|2)c(n, r)qnζr(ζ ′)r.

Thus, Fermat’s little theorem yields that

Lp−1m (φ) ≡ φ (mod p) ⇔ φ∣∣U(p) ≡ 0 (mod p).

The next proposition extends Proposition 2 of [24] to the case of Hermitian Jacobi forms of

index 1.

Proposition 5.10. If φ ∈ Jδk,1(Z(p)), then L(φ) (mod p) is the reduction of a Hermitian

Jacobi form modulo p. Moreover, I have

Ω(L(φ)

)≤ Ω(φ) + p+ 1,

with equality if and only if p 6∣∣ Ω(φ)− 1.

Proof. I proceed as in the proofs of Proposition 2 of [24] and Proposition 2.15 of [22],

and I assume that Ω(φ) = k. Recall the well known congruences Ep−1 ≡ 1 (mod p) and

Ep+1 ≡ E2 (mod p). Lemma 5.1 shows that L(φ) (mod p) ∈ ˜Jδk+p+1,1 if p ≡ 3 (mod 4) and

L(φ) (mod p) ∈ ˜J−δk+p+1,1 if p ≡ 1 (mod 4). Hence I have Ω(L(φ)

)≤ k + p+ 1.

If p divides k − 1 , then Ω(L(φ)

)≤ k + 2 < k + p + 1 by Lemma 5.1. On the other

hand, if Ω(L(φ)

)< k + p+ 1, then Ω

(k−1

3φE2

)≤ k + 2 < k + p+ 1 by Lemma 5.1. Hence

if I prove that Ω(φE2) = k + p+ 1, then this implies that p divides k − 1.

Recall that φ can be written as

φ = eφ+4,1 + fφ−6,1 + gφ

+8,1 (or φ = hφ

+,cusp10,1 ),

where e ∈ Mk−4, f ∈ Mk−6, and g ∈ Mk−8 (or h ∈ Mk−10) all have p -integral rational

coefficients by Proposition 5.5. Moreover, at least one of e, f , or g (or h) has maximal

filtration, since otherwise Ω(φ) < k. Then Theorem 2 and Lemma 5 of [29] guarantee that

either eE2, fE2, or gE2 (or hE2) has maximal filtration. I conclude that Ω(φE2) = k+p+1,

which completes the proof. �

I am now in a position to prove my main result in this section.

35

Theorem 5.11. Let φ ∈ Jδk,1(Z(p)) such that φ 6≡ 0 (mod p). If p > k , then

Ω(Lp+2−k(φ)

)=

2p+ 4− k, if φ

∣∣U(p) 6≡ 0 (mod p),p+ 5− k, if φ

∣∣U(p) ≡ 0 (mod p).Proof. I closely follow Tate’s (see §7 of [12]) original argument; see also the proofs of

Proposition 3 of [23] and Theorem 2.17 of [22]). Assume that φ|U(p) ≡ 0 (mod p). Then

Lp−1(φ) ≡ φ (mod p), and φ is in its own heat cycle. I use standard terminology and call

φ1 a low point of its heat cycle if it occurs directly after a fall, i.e., if φ1 = LA(φ) and

Ω(LA−1(φ)) ≡ 1 (mod p). Let φ1 be a low point of its heat cycle and let cj ∈ N be minimal

such that

Ω(Lcj−1(φ1)

)= Ω(φ1) + (cj − 1)(p+ 1) ≡ 1 (mod p),

and let bj ∈ N be given by

Ω(Lcj(φ1)

)= Ω(φ1) + cj(p+ 1)− bj(p− 1).

Exactly as in [12, 22, 23] one discovers that∑cj = p− 1 and

∑bj = p+ 1. I have

cj+1(p+ 1)− bj(p− 1) ≡ cj+1 + bj ≡ 0 (mod p)

and hence ∑(cj+1 + bj) =

∑cj +

∑bj = p− 1 + p+ 1 = 2p,

which shows that there is either one fall with c1 = p − 1 and b1 = p + 1 or there are two

falls with b1 = p − c2 and b2 = p − c1. There is precisely one fall if and only if Ω(φ1) = 3

(mod p). Assume now that there are two falls, and write Ω(φ1) = ap + B with 1 ≤ B ≤ p

and p 6= B − 3. In particular, if φ1 = φ then a = 0 and B = k. Hence I obtain

c1 +B − 2 ≡ 0 (mod p),

and c1 = 2−B or c1 = p+ 2−B. Note that c1 ≥ 1, and the case c1 = 2−B is only possible

if c1 = B = 1. However, this is impossible if φ1 = φ, since Jδk,1 = {0} if k < 4. For the case

36

c1 = p+ 2−B I find that

Ω(Lc1(φ1)) = (a+ 1)p+ 5−B,

and if φ1 = φ, then this gives the desired formula.

Now assume that φ|U(p) 6≡ 0 (mod p). By assumption, p > k. One finds that L(φ)

is a low point of its heat cycle (see also [23]). It’s filtration equals Ω(L(φ)

)= p+ k+ 1, i.e.,

a = 1 and B = k + 1 in my previous notation. The case c1 = 2 − B is impossible, since

c1 ≥ 1 implies k < 1. Therefore, c1 = p+ 2−B, and Lp+2−k(φ) = 2p+ 4− k. �

5.2. Examples

The following table provides all U(p) congruences with 5 ≤ p < 100 for examples

of Hermitian Jacobi forms. If a prime p is not listed, then the tables of Fourier series

coefficients in the Appendix show that there exists a coefficient c(n, r) 6≡ 0 (mod p) such

that p | 4(n − |r|2). I write p to indicate that I apply Theorem 5.11, while for the other

primes p that are listed, I verify directly that Lp−1(φ) ≡ φ (mod p) for a Hermitian Jacobi

form φ.

Table 5.1. Examples

Cusp form parity Weight U(p) congruences p < 100

φ+8,1 − E4φ+4,1 + 8 5

φ+,cusp10,1 + 10 5, 23

E6φ+4,1 − E4φ−6,1 - 10 5, 7

E4φ+8,1 − E24φ+4,1 + 12 5

E4φ+,cusp10,1 + 14 5

E6φ+,cusp10,1 - 16 5,11,13

E24φ+,cusp10,1 + 18 5, 7, 13, 23 , 79

37

The proof of the U(p) congruences in Table 5.1 relies on the identities

L(E2) =1

3(E22 − E4)

L(E4) =4

3(E2E4 − E6)

L(E6) = 2(E2E6 − E24)

(21)

and also the identities from Example 5.3:

L(φ+4,1) = E2φ+4,1 − φ−6,1

L(φ−6,1) =5

3E2φ

−6,1 −

8

3E4φ

+4,1 + φ

+8,1

L(φ+8,1) =7

3E2φ

+8,1 −

14

9E6φ

+4,1 −

7

9E4φ

−6,1

L(φ+,cusp10,1 ) = 3E2φ+,cusp10,1 .

(22)

My calculations were performed with the help of Mathematica.

Consider p = 5. Recall the congruences E4 ≡ 1 (mod 5) and E2 ≡ E6 (mod 5). If

p = 5 and φ is an element in Table 5.1, then a direct calculation shows that L4(φ) ≡ φ

(mod 5). Specifically,

L4(φ+,cusp10,1

)≡(2E22E4 + E

22 + 3E2E6

)φ+,cusp10,1 ≡ φ

+,cusp10,1 (mod 5).

Observe that φ+,cusp10,1 ≡ E4φ+,cusp10,1 ≡ E24φ

+,cusp10,1 (mod 5). Thus, if φ = φ

+,cusp10,1 , φ = E4φ

+,cusp10,1 ,

or φ = E24φ+,cusp10,1 , then

L4(φ) ≡ φ (mod 5).

Moreover,

L4(φ+8,1 − E4φ+4,1) ≡ E24(φ+8,1 + 4E4φ

+4,1

)≡ φ+8,1 − E4φ+4,1 (mod 5)

and hence

L4(E4φ+8,1 − E24φ+4,1) ≡ L4(φ+8,1 − E4φ+4,1) ≡ φ+8,1 − E4φ+4,1 ≡ E4φ+8,1 − E24φ+4,1 (mod 5).

Finally,

L4(E6φ

+,cusp10,1

)≡ E4E6φ+,cusp10,1 ≡ E6φ

+,cusp10,1 (mod 5)

38

and

L4(E6φ+4,1 − E4φ−6,1) ≡ E4

(E6φ

+4,1 + 4E4φ

−6,1

)≡ E6φ+4,1 − E4φ−6,1 (mod 5) .

Consider p = 7. Note that E6 ≡ 1 (mod 7) and E2 ≡ E8 (mod 7). Direct calcula-

tions show that

L6(E6φ+4,1 − E4φ−6,1) ≡ E26

(E6φ

+4,1 + 6E4φ

−6,1

)≡ E6φ+4,1 − E4φ−6,1 (mod 7)

and

L6(E24φ

+,cusp10,1

)≡ E24φ

+,cusp10,1 (mod 7) .

Consider p = 11. Note that E10 ≡ E4E6 ≡ 1 (mod 11) and E2 ≡ E12 ≡ 5E34 + 7E26(mod 11). A direct calculation shows that

L10(E6φ

+,cusp10,1

)≡(

9E94 + 3E54E6 + 2E

104 E6 + 4E

64E

26 + 3E

24E

36

+ 4E74E36 + 10E

34E

46 + 10E

44E

56 + 10E

66 + E4E

76

)φ+,cusp10,1

≡ (9E94 + 3E44 + 2E94 + 4E44 + 3E6

+ 4E44 + 10E6 + 10E6 + 10E66 + E

66)φ

+,cusp10,1

≡ (11E94 + 11E44 + 23E6 + 11E66)φ+,cusp10,1

≡ E6φ+,cusp10,1 (mod 11).

Consider p = 13. Note that E12 ≡ 6E34 + 8E26 ≡ 1 (mod 13) and E2 ≡ E14 ≡ E24E6

(mod 13). Direct calculations show that

L12(E6φ

+,cusp10,1

)≡ E6

(10E64 + 5E

34E

26 + 12E

46

)φ+,cusp10,1 ≡ E6E212φ

+,cusp10,1 ≡ E6φ

+,cusp10,1 (mod 13)

and

L12(E24φ

+,cusp10,1

)≡ E24

(10E64 + 9E

34E

26 + 6E

46 + 8E

94E

26 + 11E

64E

46

+ 12E34E66 + 5E

64E

26 + 9E

34E

46 + 9E

66

)φ+,cusp10,1

≡ E24E12(8E94 + 6E64E26 + 8E34E46 + 5E66)φ+,cusp10,1

39

≡ E24E212(10E64 + 5E34E26 + 12E46)φ+,cusp10,1

≡ E24E312(6E34 + 8E26)φ+,cusp10,1

≡ E24E412φ+,cusp10,1

≡ E24φ+,cusp10,1 (mod 13) .

Consider p = 23. I apply Theorem 5.11 to verify the U(p) congruences for φ+,cusp10,1 and

E24φ+,cusp10,1 . One finds that E22 ≡ 10E44E6 + 14E4E36 ≡ 1 (mod 23).

Let φ = φ+,cusp10,1 . Observe that p+ 2− k = 15. A direct calculation shows that

L15(φ+,cusp10,1

)≡ 12E24

(10E44E6 + 14E4E

36

)φ+,cusp10,1 ≡ 12E24E22φ

+,cusp10,1 ≡ 12E24φ

+,cusp10,1 (mod 23).

Hence Ω(L23+2−10

(φ+,cusp10,1

))= 18 = 23 + 5− 10, and φ+,cusp10,1 |U(23) ≡ 0 (mod 23).

Let φ = E24φ+,cusp10,1 . Observe that p+ 2− k = 10. A direct calculation shows that

L7(E24φ

+,cusp10,1

)≡(20E44E6 + 5E4E

36

)φ+,cusp10,1 ≡ 2E22φ

+,cusp10,1 ≡ 2φ

+,cusp10,1 (mod 23).

Hence Ω(L23+2−18

(E24φ

+,cusp10,1

))= 10 = 23+5−18, and E24φ

+,cusp10,1 |U(23) ≡ 0 (mod 23).

Consider p = 79. I apply Theorem 5.11 to verify the U(p) congruences for E24φ+,cusp10,1 .

One finds that

E78 ≡ 26E184 E6+10E154 E36 +73E124 E56 +33E94E76 +41E64E96 +72E34E116 +62E136 ≡ 1 (mod 79).

A direct calculation shows that

L63(E24φ

+,cusp10,1

)≡(

73E324 E6 + 46E294 E

36 + 70E

264 E

56 + 12E

234 E

76 + 57E

204 E

96 + 75E

174 E

116

+ 61E144 E136 + 9E

114 E

156 + 16E

84E

176 + 39E

54E

196 + 31E

24E

216

)φ+,cusp10,1

≡ E78(18E144 + 7E

114 E

26 + 71E

84E

46 + 37E

54E

66 + 40E

24E

86

)φ+,cusp10,1

≡(18E144 + 7E

114 E

26 + 71E

84E

46 + 37E

54E

66 + 40E

24E

86

)φ+,cusp10,1 (mod 79).

Hence Ω(L79+2−18

(E24φ

+,cusp10,1

))= 66 = 79 + 5− 18, and E24φ

+,cusp10,1 |U(79) ≡ 0 (mod 79).

40

APPENDIX

FOURIER COEFFICIENTS OF HERMITIAN JACOBI FORMS

41

In this appendix, I give three tables of Fourier series coefficients of Hermitian Jacobi

forms. Table A.1 contains the Fourier series coefficients of the generators φ+4,1, φ−6,1, and φ

+8,1,

which are not cusp forms. Table A.2 and Table A.3 contain Fourier series coefficients of cusp

forms of weights 8, 10, 12, and 10, 14, 16, 18, respectively. The Fourier series coefficients in

Tables A.2 and A.3 imply the non-existence of U(p) congruences in Table 5.1 of Chapter 5.

Let φδ =∑c(n, r)qnζrζ ′ ∈ Jk,1(Z(p)). Recall that Proposition 4.6 implies that c(n, r)

depends only on n− |r|2 and r (mod O). Set D := 4(n− |r|2).

If φδk,1 is not generated by φ+,cusp10,1 (such as the forms in Tables A.1 and A.2), then

c(n, r) depends only on D, and I write c(D) := c(n, r). In particular, if D ≡ 1 (mod 4),

then c(D) = 0.

On the other hand, if φδ is generated by the cusp form φ+,cusp10,1 (such as the forms

in Table A.3), then Proposition 4.6, Lemma 4.9, and Theorem 4.14 imply that c(n, r) =

−c(n, ir), and if D 6≡ 3 (mod 4), then c(n, r) = 0 for every r. If D ≡ 3 (mod 4), then the

cases 2

Table A.1. Fourier coefficients of non-cusp forms

coeff. φ+4,1 φ−6,1 φ

+8,1

c(0) 1 1 1

c(2) 12 -12 28

c(3) 32 -64 0

c(4) 60 -204 364

c(6) 160 -1088 2912

c(7) 192 -1920 8192

c(8) 252 -3276 16044

c(10) 312 -7512 64792

c(11) 480 -11712 114688

c(12) 544 -16448 200928

c(14) 960 -32640 503360

c(15) 832 -40064 745472

c(16) 1020 -52428 1089452

c(18) 876 -77772 2186940

c(19) 1440 -104256 3096576

c(20) 1560 -127704 4196920

c(22) 2400 -199104 7544992

c(23) 2112 -223872 9691136

c(24) 2080 -262208 12547808

c(26) 2040 -342744 19975256

c(27) 2624 -419968 25346048

c(28) 3264 -493440 31553344

c(30) 4160 -681088 48484800

c(31) 3840 -738816 58261504

c(32) 4092 -838860 70439852

c(34) 3480 -1002264 99602104

43

coeff. φ+4,1 φ−6,1 φ

+8,1

c(35) 4992 -1201920 120553472

c(36) 4380 -1322124 142487436

c(38) 7200 -1772352 200569824

c(39) 5440 -1827968 230350848

c(40) 6552 -2050776 268594872

c(42) 4608 -2304000 354052608

c(43) 7392 -2735040 414482432

c(44) 8160 -3009984 476105504

c(46) 10560 -3805824 630908096

c(47) 8832 -3903744 706822144

c(48) 8224 -4194368 800698080

c(50) 7812 -4695012 1008274932

c(51) 9280 -5345408 1152499712

c(52) 10200 -5826648 1296257144

c(54) 13120 -7139456 1648943296

c(55) 12480 -7331712 1815224320

c(56) 12480 -7866240 2022013760

c(58) 10104 -8487384 2457911512

c(59) 13920 -9693888 2765815808

c(60) 14144 -10296448 3056208064

c(62) 19200 -12559872 3783060736

c(63) 14016 -12443520 4094140416

c(64) 16380 -13421772 4507001772

c(66) 11520 -14054400 5327212800

44

coeff. φ+4,1 φ−6,1 φ

+8,1

c(67) 17952 -16120896 5931089920

c(68) 17400 -17038488 6481076056

c(70) 24960 -20432640 7835684480

c(71) 20160 -20329344 8400838656

c(72) 18396 -21231756 9123064524

c(74) 16440 -22489944 10599441944

c(75) 20832 -25040064 11656200192

c(76) 24480 -26793792 12637846368

c(78) 27200 -31075456 14977074112

c(79) 24960 -31160064 15939682304

c(80) 26520 -32819928 17190762680

c(82) 20184 -33909144 19624082296

c(83) 27552 -37966656 21438398464

c(84) 23040 -39168000 22999441920

c(86) 36960 -46495680 26943381920

c(87) 26944 -45266048 28394283008

c(88) 31200 -47984064 30453867808

c(90) 22776 -48685272 34259226456

c(91) 32640 -54839040 37237719040

c(92) 35904 -57535104 39772497856

c(94) 44160 -66363648 45944149888

c(95) 37440 -65264256 48205946880

c(96) 32800 -67108928 51254988512

c(98) 28236 -69148812 57180430300

c(99) 35040 -75905472 61652140032

c(100) 39060 -79815204 65560474980

45

coeff. φ+4,1 φ−6,1 φ

+8,1

c(102) 46400 -90871936 74898602688

c(103) 42432 -90040704 78300651520

c(104) 42840 -93569112 82969759992

c(106) 33720 -94685784 91559642776

c(107) 45792 -104863680 98406776832

c(108) 44608 -107931776 103942820800

c(110) 62400 -124639104 117990483520

c(111) 43840 -119946368 122481795072

c(112) 49344 -125831040 129431980864

c(114) 34560 -125107200 141494895360

c(115) 54912 -140143872 151685660672

c(116) 50520 -144285528 159727123192

c(118) 69600 -164796096 179785397792

c(119) 55680 -160362240 186209976320

c(120) 54080 -164142208 195547235520

c(122) 44664 -166150104 212838007256

c(123) 53824 -180848768 226757672960

c(124) 65280 -189875712 238425727232

c(126) 70080 -211539840 266125853760

c(127) 64512 -208115712 275141787648

c(128) 65532 -214748364 288397763500

c(130) 53040 -214557744 311586919280

c(131) 68640 -235599936 331408523264

c(132) 57600 -238924800 346312162560

c(134) 89760 -274055232 385557888352

46

coeff. φ+4,1 φ−6,1 φ

+8,1

c(135) 68224 -262899968 396428591104

c(136) 73080 -273618072 414923166168

c(138) 50688 -268646400 445206547968

c(139) 77280 -298640832 472954273792

c(140) 84864 -308893440 493887481984

c(142) 100800 -345598848 546004490304

c(143) 81600 -334518144 560718061568

c(144) 74460 -339785868 583863854028

c(146) 63960 -340778904 625190905976

c(147) 75296 -368793664 660744634368

c(148) 82200 -382329048 688960823096

c(150) 104160 -425681088 757607803680

c(151) 91200 -415908480 777297059840

c(152) 93600 -427136832 808702798176

c(154) 69120 -421632000 861022901760

c(155) 99840 -462498816 909386383360

c(156) 92480 -469787776 944041984704

c(158) 124800 -529721088 1036108551296

c(159) 89920 -504990848 1058084937728

c(160) 106392 -525126360 1100216025272

c(162) 70860 -510261132 1165144355676

c(163) 106272 -564729408 1229857456128

c(164) 100920 -576455448 1275522598168

c(166) 137760 -645433152 1393515087328

c(167) 111552 -622237056 1422435983360

47

coeff. φ+4,1 φ−6,1 φ

+8,1

c(168) 96768 -628992000 1472248447488

c(170) 90480 -627417264 1558158936880

c(171) 105120 -675683136 1637234049024

c(172) 125664 -702905280 1698274168864

c(174) 134720 -769522816 1845702645696

c(175) 124992 -751201920 1883566006272

c(176) 123360 -767569344 1948967103776

c(178) 95064 -752906904 2053110455032

c(179) 128160 -821300544 2156987449344

c(180) 113880 -827649624 2226853083000

c(182) 163200 -932263680 2420409047680

c(183) 119104 -886133888 2459464949760

c(184) 137280 -917203584 2544724005824

c(186) 92160 -886579200 2669176473600

c(187) 139200 -978209664 2804017119232

c(188) 150144 -1003262208 2895899350400

c(190) 187200 -1109492352 3133379471040

c(191) 145920 -1064690688 3183669379072

c(192) 131104 -1073741888 3280511471328

c(194) 112920 -1062351384 3441095470904

c(195) 141440 -1144307968 3600555212800

c(196) 141180 -1175529804 3716696448556

c(198) 175200 -1290393024 4007368249248

c(199) 158400 -1254591360 4072390533120

c(200) 164052 -1281738276 4196991695652

48

Table A.2. Fourier coefficients of cusp forms (I)

coeff. 116(φ+8,1 − E4φ

+4,1)

124(E6φ

+4,1 − E4φ

−6,1)

116(E4φ

+8,1 − E24φ

+4,1)

c(2) 1 1 1

c(3) -2 4 -2

c(4) 4 -20 4

c(6) -8 -80 232

c(7) 20 56 -460

c(8) -48 144 912

c10) 10 610 250

c(11) -62 -740 418

c(12) 224 -448 -2656

c(14) 80 -1120 -8080

c(15) -20 2440 14860

c(16) -448 2240 -23488

c(18) -231 -3423 4329

c(19) 486 -780 -38874

c(20) 40 -12200 123880

c(22) -248 14800 74392

c(23) -676 -9496 -53956

c(24) 1408 29440 -171392

c(26) 1466 -5470 -25654

c(27) -996 12552 149724

c(28) -2240 -6272 126400

c(30) -80 -48800 -433520

c(31) 2704 -2720 388624

c(32) 1280 -81664 -874240

c(34) -4766 73090 -67166

49

coeff. 116(φ+8,1 − E4φ

+4,1)

124(E6φ

+4,1 − E4φ

−6,1)

116(E4φ

+8,1 − E24φ

+4,1)

c(35) 200 34160 -514360

c(36) -924 68460 2229156

c(38) 1944 15600 1928904

c(39) -2932 -21880 -1837972

c(40) -480 87840 -1246560

c(42) 9600 -139776 685440

c(43) -1390 -237316 2132210

c(44) 6944 82880 -919456

c(46) -2704 189920 -5476144

c(47) -488 305296 1903192

c(48) -8704 -474112 3570176

c(50) -15525 -18525 -1838805

c(51) 9532 292360 -449348

c(52) 5864 109400 -15216856

c(54) -3984 -251040 5856336

c(55) -620 -451400 4758580

c(56) -14080 412160 18579200

c(58) 25498 -128222 4394218

c(59) -5062 -149140 -18650662

c(60) 2240 -273280 4251200

c(62) 10816 54400 13410496

c(63) -4620 -191688 -9179820

c(64) 33792 1059840 -3406848

c(66) -29760 1847040 -15105600

c(67) -25442 610756 38524318

50

coeff. 116(φ+8,1 − E4φ

+4,1)

124(E6φ

+4,1 − E4φ

−6,1)

116(E4φ

+8,1 − E24φ

+4,1)

c(68) -19064 -1461800 -4938104

c(70) 800 -683200 -52155040

c(71) 34356 -47880 2756436

c(72) 11088 -492912 -22594032

c(74) 1994 -3472030 38853434

c(75) 31050 -74100 -8303190

c(76) -54432 87360 -20658912

c(78) -11728 437600 54505232

c(79) -20056 1437680 15125864

c(80) -4480 1366400 3958400

c(82) 29362 2146882 -71679278

c(83) -13178 -2080076 -54031418

c(84) 38400 2795520 157570560

c(86) -5560 4746320 23736920

c(87) -50996 -512888 -83606996

c(88) 43648 -5446400 -80177792

c(90) -2310 -2088030 142640010

c(91) 29320 -306320 60922120

c(92) 75712 1063552 69166912

c(94) -1952 -6105920 -85725152

c(95) 4860 -475800 123683100

c(96) -124928 1945600 -242505728

c(98) 21649 3807937 -282407951

c(99) 14322 2533020 85582962

c(100) -62100 370500 -103201620

51

coeff. 116(φ+8,1 − E4φ

+4,1)

124(E6φ

+4,1 − E4φ

−6,1)

116(E4φ

+8,1 − E24φ

+4,1)

c(102) 38128 -5847200 3095248

c(103) 89668 4183384 -32936732

c(104) -70368 -787680 157974432

c(106) -192854 824290 405220186

c(107) -74190 4016316 -176750190

c(108) 111552 -1405824 -79858368

c(110) -2480 9028000 -4503440

c(111) -3988 -13888120 48321932

c(112) 87040 -6637568 832936960

c(114) 233280 1946880 -506208960

c(115) -6760 -5792560 -350149480

c(116) 101992 2564440 -580971608

c(118) -20248 2982800 549816632

c(119) -95320 4093040 46072040

c(120) 14080 17958400 55456000

c(122) -10918 -14746078 749999402

c(123) -58724 8587528 464136796

c(124) -302848 304640 -185797888

c(126) -18480 3833760 -937409040

c(127) 108096 10294656 -457765824

c(128) 53248 -290816 -570847232

c(130) 14660 -3336700 -973724860

c(131) 183614 -12497020 1074684734

c(132) -119040 -36940800 185091840

c(134) -101768 -12215120 -483208088

52

coeff. 116(φ+8,1 − E4φ

+4,1)

124(E6φ

+4,1 − E4φ

−6,1)

116(E4φ

+8,1 − E24φ

+4,1)

c(135) -9960 7656720 373907160

c(136) 228768 10524960 -5222112

c(138) -324480 23702016 1058532480

c(139) 8962 -11810660 -1041206558

c(140) -22400 -3825920 658779520

c(142) 137424 957600 1955683824

c(143) -90892 4047800 -583175212

c(144) 103488 -7667520 75267648

c(146) 288626 -5725630 -1306972654

c(147) -43298 15231748 -1383446498

c(148) 7976 69440600 1163644136

c(150) 124200 1482000 -43217160

c(151) -361060 -33506200 1036362620

c(152) -342144 -5740800 -895373184

c(154) 297600 25858560 1301811840

c(155) 27040 -1659200 855940000

c(156) 328384 2450560 -536432576

c(158) -80224 -28753600 -512160544

c(159) 385708 3297160 289412428

c(160) 12800 -49815040 -1752102400

c(162) -646479 -53788095 -664381359

c(163) -119154 23158884 713350926

c(164) 117448 -42937640 2279508808

c(166) -52712 41601520 -4453389752

c(167) 514148 18739736 -776285692

53

coeff. 116(φ+8,1 − E4φ

+4,1)

124(E6φ

+4,1 − E4φ

−6,1)

116(E4φ

+8,1 − E24φ

+4,1)

c(168) -460800 -20127744 -1232824320

c(170) -47660 44584900 -315635660

c(171) -112266 2669940 2232943254

c(172) 155680 26579392 -421304480

c(174) -203984 10257760 3413652496

c(175) -310500 -1037400 1157351100

c(176) -269824 87710720 -769759744

c(178) 310738 -83324222 1640018098

c(179) -150666 5680980 -4652889546

c(180) -9240 41760600 -1175687640

c(182) 117280 6126400 6373894240

c(183) 21836 -58984312 -3972513364

c(184) 475904 -69890560 6454064384

c(186) 1297920 6789120 -2001277440

c(187) 295492 -54086600 -204930428

c(188) 54656 -34193152 906897536

c(190) 19440 9516000 -6429036720

c(191) -736928 39730240 2392024672

c(192) 57344 82460672 7572660224

c(194) -1457086 120619010 3115119554

c(195) -29320 -13346800 3499362680

c(196) 86596 -76158740 -16715731004

c(198) 57288 -50660400 -93782952

c(199) 316020 99694200 3688094100

c(200) 745200 -2667600 -526833360

54

Table A.3. Fourier coefficients of cusp forms (II)

coeff. φ+,cusp10,1 E4φ+,cusp10,1 E6φ

+,cusp10,1 E

24φ

+,cusp10,1

c(3) 1 1 1 1

c(7) -18 222 -522 462

c(11) 135 -2025 -7425 53415

c(15) -510 -270 107850 -30

c(19) 765 66525 -306675 -2862915

c(23) 1242 -294678 -490158 5399802

c(27) -7038 397602 1743858 42850242

c(31) 8280 283800 12396600 -146205480

c(35) 9180 -59940 -56297700 -13860

c(39) -27710 -4386350 59275450 -37084190

c(43) 3519 3193359 35870679 2540495199

c(47) 20196 21555396 1583604 -4768583004

c(51) 50370 -39383550 -44896350 -1382996670

c(55) -68850 546750 -800786250 -1602450

c(59) -153765 30494475 1130436675 22485110715

c(63) 244782 -29712258 1586415942 -90773298

c(67) 52785 149495985 -1493137935 -56276126415

c(71) -71010 -152996850 -5611095450 28887370110

c(75) -130525 -244067725 5528106875 -152587889725

c(79) -343620 149070300 2995497900 376472333820

c(83) 517293 487109133 -8806141827 182370961773

c(87) 54978 163560978 30520168602 -722997605022

c(91) 498780 -973769700 -30941784900 -17132895780

c(95) -390150 -17961750 -33074898750 85887450

c(99) -1835865 271023975 22565399175 -10494925785

55

coeff. φ+,cusp10,1 E4φ+,cusp10,1 E6φ

+,cusp10,1 E

24φ

+,cusp10,1

c(103) 1161270 1661670 28336683870 1116120818070

c(107) 896751 -392717889 100697615559 2642126595471

c(111) 793730 3600024050 -125971660150 -5014816412830

c(115) -633420 79563060 -52863540300 -161994060

c(119) -906660 -8743148100 23435894700 -638944461540

c(123) -75582 2124864738 -41444675982 -6629976268542

c(127) -2589984 1122288096 91058573664 16248773450976

c(131) 1523745 9101793825 8610005025 6112002243105

c(135) 3589380 -107352540 188075085300 -1285507260

c(139) 2472615 -9534945225 312508108575 -30682651761465

c(143) -3740850 8882358750 -440120216250 -1980852008850

c(147) -767039 -18946372319 -1453168573871 24002928988801

c(151) -4649670 10144831050 732120620850 -26706876606630

c(155) -4222800 -76626000 1336973310000 4386164400

c(159) 11166210 -13585251150 473374435050 74221986126690

c(163) 1718937 36420494937 -737696115783 68592359731737

c(167) 4728294 16254954774 -507091281714 -159398150972346

c(171) -10403235 -8903639475 932019365925 562502676285

c(175) 2349450 -54183034950 -2885671788750 -70495605052950

c(179) 5331285 -473303472