HERMITIAN JACOBI FORMS AND CONGRUENCES
Jayantha Senadheera
Dissertation Prepared for the Degree of
DOCTOR OF PHILOSOPHY
UNIVERSITY OF NORTH TEXAS
August 2014
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 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. Copyright 2014
by
Jayantha Senadheera
ii 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.
iii 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
iv 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
n a b o Γ := SL2(Z) = ( c 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. aτ + b (ii) f = (cτ + d)kf(τ) ∀ ( a b ) ∈ Γ. cτ + d c d (iii) The Fourier series expansion of f is of the form
∞ X f(τ) = c(n)e2πinτ . n=0
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
n a b o Γ0(4) := ( c 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
∞ 2k X E (τ) := 1 − σ (n)qn, k B k−1 k n=1 where
X k−1 σk−1(n) := d d|n
is the usual divisor function, and where the Bernoulli numbers Bk are defined by ∞ x X xk = B . ex − 1 k k! k=0
If k ≥ 4, then Ek ∈ Mk. If k = 2, then E2 is a so-called quasimodular form, which satisfies aτ + b 6c (1) E = (cτ + d)2 E (τ) + (cτ + d) ∀ ( a b ) ∈ Γ . 2 cτ + d 2 πi c d
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 ∆ := (E3 − E2) ∈ S . 1728 4 6 12
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
X πi(a+n)2τ+2πi(n+a)(z+b) (2) θa,b(τ, z) = e . n∈Z Of particular interest are the following special cases:
P πin2τ+2πinz (i) θ0,0(τ, z) = e P πin2τ+2πin(z+ 1 ) (ii) θ 1 (τ, z) = e 2 0, 2 P πi(n+ 1 )2τ+2πi(n+ 1 )z (iii) θ 1 (τ, z) = e 2 2 2 ,0
4 P πi(n+ 1 )2τ+2πi(n+ 1 )(z+ 1 ) (iv) θ 1 1 (τ, z) = e 2 2 2 . 2 , 2
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 n2 x := θ0,0(τ, 0) = 1 + 2 q 2 n=1 ∞ 2 X n n (3) y := θ 1 (τ, 0) = 1 + 2 (−1) q 2 0, 2 n=1 ∞ 1 X n(n+1) z := θ 1 (τ, 0) = 2q 8 q 2 . 2 ,0 n=0
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
1 8 8 8 (ii) E4 = 2 (x + y + z ) 1 4 4 4 4 4 4 (iii) E6 = 2 (x + y )(x + z )(y − z ) (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
0 if k < 0 or if k is odd, dim M = k k 12 if k ≥ 0 k ≡ 2 (mod 12), k 12 + 1 otherwise.
In particular, if f ∈ MK is nonconstant, then k ≥ 4 and k is even. Finally, if k ≥ 4, then
dim Sk = dim Mk − 1.
5 2.2. U(p) Congruences of Modular Forms
P∞ n Let f(τ) = n=0 a(n)q be a modular form, and let d be a positive integer. Then Atkin’s U-operator is defined by
∞ X n f U(d) := a(nd)q . n=0 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).
d 1 d The Ramanujan theta operator Θ := q dq = 2πi dτ acts on the Fourier series of f by
∞ ∞ X X Θ a(n)qn = na(n)qn. n=0 n=0
Remark 2.6. If f ∈ Mk, then
k (4) Θ(f) = fE + f,ˆ 12 2 ˆ where f ∈ Mk+2. In particular, if f = E4 or f = E6, then one finds Ramanujan’s [21] identities 1 1 Θ(E ) = (E E − E ) and Θ(E ) = (E E − E ). 4 3 4 2 6 6 2 6 2 8
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
Mfk := f (mod p): f(τ) ∈ Mk ∩ Z[[q]] , and let ω(f) := inf k : f (mod p) ∈ Mfk 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 2p − k + 2, if f U(p) 6≡ 0 (mod p); ω Θp−k+1(f) = 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). n o 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 0,X0) := (MM 0,XM 0 + X0) and this group is called the full Jacobi group.
2 Next I define actions of the groups SL2(Z) and Z on functions φ : H × C → C, the so-called slash operators.
Definition 3.2. Let φ : H × C → C. Then 2πimcz2 a b −k − cτ+d aτ+b z a b (i) φ|k,m [ c d ] (τ, z) := (cτ + d) e φ cτ+d , cτ+d , ∀ [ c d ] ∈ Γ 2πm(λ2τ+2λz) 2 (ii) (φ|m[λ, µ])(τ, z) := e φ(τ, z + λτ + µ), (∀ (λ, µ) ∈ Z )
If M,M 0 ∈ Γ and X,X0 ∈ Z2, then one easily verifies the relations
0 0 (φ|k,mM)|k,mM = φ|k,m(MM ),
0 0 (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,
2 (ii) φ|mX = φ ∀ X ∈ Z , and φ has a Fourier series expansion of the form (iii) ∞ X X φ(τ, z) = c(n, r)qnζr, n=0 r∈Z 4nm−r2≥0 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
2 4mn − r > 0. I denote the space of Jacobi forms of weight k and index m by Jk,m, and the cusp space of cusp form in Jk,m by Jk,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
X J n 1 n o Ek,m(τ, z) := 1|k,m, where Γ∞ := ± [ 0 1 ] , (0, µ) |n, µ ∈ Z . J J γ∈Γ∞\Γ
Then Ek,m ∈ Jk,m and one finds that
2 1 X X −k 2πim λ2 aτ+b +2λ z − cz E (τ, z) = (cτ + d) e cτ+d cτ+d cτ+d . k,m 2 c,d∈Z λ∈Z (c,d)=1
Let en,r denote the Fourier series coefficients of Ek,m. In the special case that m = 1 one H(k−1,4n−r2) can show that ek,1(n, r) = ζ(3−2k) , where H(k − 1,N) is Cohen’s function [3]. Using the values of Cohen’s function one obtains:
2 −1 −2 2 −1 −2 2 E4,1 = 1 + (ζ + 56ζ + 126 + 56ζ + ζ )q + (126ζ + 576ζ + 756 + 576ζ + 126ζ )q + ...
2 −1 −2 2 −1 −2 2 E6,1 = 1+(ζ −88ζ −330−88ζ +ζ )q+(−330ζ −4224ζ −7524−4224ζ −330ζ )q +...
9 One can use the Jacobi-Eisenstein series to define the two important cusp forms 1 φ = (E E − E E ) ∈ J cusp, 10,1 144 6 4,1 4 6,1 10,1 1 φ = (E2E − E E ) ∈ J cusp, 12,1 144 4 4,1 6 6,1 12,1 which have the following Fourier series expansions:
−1 2 −1 −2 2 φ10,1 = (ζ − 2 + ζ )q + (−2ζ − 16ζ + 36 − 16ζ − 2ζ )q + ...
−1 2 −1 −2 2 φ12,1 = (ζ + 10 + ζ )q + (10ζ − 88ζ − 132 − 88ζ + 10ζ )q + ...
Note that φ12,1 is a meromorphic Jacobi form of weight 2 and index 0. In fact, one finds that φ10,1
−1 φ12,1(τ, z) ζ + 10 + ζ −1 = −1 + 12(ζ − 2 + ζ )q + ... φ10,1(τ, z) ζ − 2 + ζ
−3 is π2 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 P 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: ∞ X ν φ(τ, z) = χν(τ)z . ν=0
Then χ0 is a modular form of weight k, but χν is not modular if ν > 0. However, one can
2πim 0 use derivatives of the χν to construct modular forms. For example ζ2 := χ2 − k χ0 is a modular form of weight k + 2. This idea extends, and one can prove that
µ X (−2πim) (k + ν − µ − 2)! (µ) ζν(τ) := χν−2µ(τ) ν (k + 2ν − 2)!µ! 0≤µ≤ 2
10 and (2πi)−ν(k + 2ν − 2)!(2ν)! D φ(τ, z) := ζ (τ) ν (k + ν − 2)! 2ν 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:
2m D := ⊕ Dν → Mk(Γ) ⊕ Sk+1(Γ) ⊕ ... ⊕ Sν(Γ). ν=0
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
cusp Mk−10 ⊕ Mk−12 → Jk,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.,
2 0 c(n, r) = c(4nm − r ), cr0 (N) = cr(N) for r ≡ r (mod 2m). This leads to the theta decomposition of a Jacobi form.
P n n Theorem 3.10. Let φ = n,r c(n, r)q ζ ∈ Jk,m. Then X φ(τ, z) = hµ(τ)θm,µ(τ, z), µ (mod 2m) where
∞ X N + r2 h (τ) := c (N)qN/4m with c (N) = c , r , (any r ∈ , r ≡ µ (mod 2m)), µ µ µ 4m Z N=0
2 cµ(N) = 0 if N 6≡ −µ (mod 4m),
and
X r2/4m r θm,µ(τ, z) := q ζ . r∈Z r≡µ (mod 2m) 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 X φ(τ, z) = hµ(τ)θ1,µ(τ, z) = h0(τ)θ1,0(τ, z) + h1(τ)θ1,1(τ, z), µ (mod 2)
and it is easy to verify that h(τ) := h0(4τ) + h1(4τ) satisfies the following transformation formulas:
(i) h(τ + 1) = h(τ).
τ 1 k− 2 (ii) h 4τ+1 = (4τ + 1) h(τ).
1 Thus, one finds that h is a modular form of weight k − 2 on the congruence subgroup Γ0(4).
Let M 1 (Γ0(4)) denote the vector space of such forms and let k− 2 n ∞ o + X N M 1 (Γ0(4)) := h ∈ Mk− 1 (Γ0(4)) h = c(N)q k− 2 2 N=0 (−1)k−1≡0,1 (mod 4)
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 1 (Γ0(4)) = Jk,1(Γ) k− 2 where the isomorphism is given by
X X a(N)qN 7−→ a(4n − r2)qnζr. N≥0 n,r∈Z N≡0,3 (mod 4) 4n−r2≥0
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.
n o Definition 4.1. The set ΓJ (O) := Γ(O) n O2 = (M, X)|M ∈ Γ(O),X ∈ O2 forms a group under the group law (M, X)(0M 0,X0) := (0MM 0,X(0M 0) + X0) 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)
−1 −k −k − 2πimczw z w φ| (M) (τ, z, w) := σ() (cτ + d) e cτ+d φ Mτ, , ∀M ∈ Γ(O) k,m,δ cτ + d cτ + d
(ii)
2πim(λλτ+zλ+λw) 2 φ|m[λ, µ] (τ, z, w) := e φ(τ, z + λτ + µ, w + λτ + µ) ∀[λ, µ] ∈ O ,
where, here and throughout δ = + if σ() = 1 and δ = − if σ() = 2.
If M, 0M 0 ∈ Γ(O) and X,X0 ∈ O2, then one can verify the relations
0 0 0 0 (φ|k,m,δ M)|k,m,δ M = φ|k,m,δ(M M ),
0 0 (φ|mX)|mX = φ|m(X + X ), and
(φ|k,m,δ M)|mXM = (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:
δ (i) φ |k,m,δ(M) = φ ∀ M ∈ Γ(O),
δ 2 (ii) φ |m[λ, µ] = φ ∀ [λ, µ] ∈ O , (iii) and φδ has a Fourier series expansion of the form
∞ X X φδ(τ, z, w) = c(n, r)qnζr(ζ0)r, n=0 r∈O# nm−|r|2≥0
2πiτ 2πiz 0 2πiw # i where as before q := e and ζ := e , and also ζ := e and O := 2 O 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,
δ δ δ,cusp and of parity δ by Jk,m(O), and the space of cusp form in Jk,m(O) by Jk,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 Jk,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
+ − n + − + + − − o Jk,m(O) := Jk,m(O) ⊕ Jk,m(O) = (φ , φ ) | φ ∈ Jk,m(O), φ ∈ Jk,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 P c(n, r)qnζr(ζ0)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 ≡ r0 (mod mO), and nm − |r|2 = n0m − |r0|2. If r0 = r + mλ (λ ∈ O), then
n0m − |r0|2 = n0m − |r|2 − m(λr + λr) − m2|λ|2 = nm − |r|2 implies that n0m = nm + mλr + mλr + m2|λ|2, i.e., n0 = n + λr + λr + m|λ|2. The elliptic transformation law (see (ii) of Definition 4.3) with λ = 0 yields
X 2 X c(n, r)qnζr(ζ0)r = qm|λ| ζλm(ζ0)λm c(n, r)qn+λr+λrζr(ζ0)re2πi(µr+µr)
16 X 2 = c(n, r)qn+λr+λr+m|λ| ζλm+r(ζ0)λm+r.
Thus, I find that c(n, r) = c(n + λr + λr + m|λ|2, r + λm) = c(n0, r0), and (i) is proved.
0 The modular transformation law (see (i) of Definition 4.3) with M = ( 0 ) asserts that
φδ(τ, z, −1w) = σ()kφδ(τ, z, w),
which implies that
X X c(n, r)qnζr(ζ0)r = σ()k c(n, r)qnζr(ζ0)r.
Hence σ()kc(n, r) = c(n, r), and (ii) holds.
If m = 1 and n−|r2| = n0 −|r0|2 then r ≡ r0 (mod O) or r ≡ ir0 (mod O). In the first case, c(n, r) = c(n0, r0) by (i), and in the second case, c(n, r) = c(n0, ir0) = σ(−i)(−i)kc(n0, r0).
Therefore, if k ≡ 0 (mod 4) and δ = +, then c(n, r) = c(n0, r0) and if k ≡ 2 (mod 4) and δ = −, then c(n, r) = c(n0, r0). I conclude that (iii) and (iv) hold.
Finally, if m = 1 and k is odd, then the modular transformation law with = 1 and
−1 0 2 2 M = 0 −1 guarantees that c(n, r) = −c(n, −r). Note that n − |r| = n − | − r| 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 = P c(n, r)qnζr(ζ0)r.
# Then for s ∈ O /mO I define functions cs : Z → C by N+4|r|2 2 c 4m , r if N ≡ −4|s| (mod 4m) cs(N) = 0 otherwise,
where r ∈ O#, r ≡ s (mod mO). Moreover, set
∞ X N/4m # hs(τ) := cs(N)q (r ∈ O , r ≡ s (mod mO))(5) N=0
17 and
H X |r|2/m r 0 r θm,s(τ, z, w) := q ζ (ζ ) . r∈O# r≡s (mod mO) δ P n r 0 r δ Exactly as in Haverkamp [9] one verifies that φ = n,r c(n, r)q ζ (ζ ) ∈ Jk,m(O) has the theta decomposition:
δ X H (6) φ (τ, z, w) = hs(τ)θm,s(τ, z, w). s∈O#/mO
H Haverkamp [9] determines the transformation laws of the theta functions θm,s. The following proposition is Corollary 4.4 of [9].
1 0 1 1 0 1 Proposition 4.7. Set I = ( 0 1 ) ,T = ( 0 1 ) , and J = ( −1 0 ). Then
H H θm,s m[λ, µ] = θm,s for λ, µ ∈ O,
H H × θm,s 1,m,+I = θm,s for ∈ O ,
2πi|s|2 H m H θm,s 1,m,+T = e θm,s,
H i X 4πiRe(st) H θ J = e m θ . m,s 1,m,+ 2m m,t t∈O#/mO
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.
δ δ Corollary 4.8. Let φ ∈ Jk,m(O) and hs as in (6). Then hs(τ) if δ = + k i h−is(τ) = −hs(τ) if δ = −
2 − 2πi|s| hs(τ + 1) = e m hs(τ).
i 0 Proof. The modular transformation law (see (i) of Definition 4.3) with M = ( 0 i ) implies that
(7) φδ(τ, iz, −iw) = σ(i)ikφδ(τ, z, w).
18 H Note that the second transformation of Proposition 4.7 asserts that θm,s(τ, iz, −iw) = H θm,−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
1 i 1+i # the set {0, 2 , 2 , 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.
δ δ Lemma 4.9. Let φ ∈ Jk,1(O) and hs as in (6).
If k ≡ 0 (mod 4) and δ = +, then h i (τ) = h 1 (τ). 2 2
If k ≡ 2 (mod 4) and δ = +, then h0(τ) = h 1+i (τ) = 0, h i (τ) = −h 1 (τ). 2 2 2
If k ≡ 0 (mod 4) and δ = −, then h0(τ) = h i+1 (τ) = 0, h i (τ) = −h 1 (τ). 2 2 2
If k ≡ 2 (mod 4) and δ = −, then h i (τ) = h 1 (τ). 2 2
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.
+ + +, cusp +, cusp The Hermitian Jacobi forms φk,1 ∈ Jk,1(O) for k = 4, 8, 12, 16 and φ10,1 ∈ J10,1 (O)
19 were considered in [25] and it was shown that
+ 1 6 6 H 1 6 H H 1 6 6 H φ4,1 = (x + y )θ1,0 + z (θ1, 1 + θ1, i ) + (x − y )θ1, 1+i 2 2 2 2 2 2
+ 1 14 14 H 1 14 H H 1 14 14 H φ8,1 = (x + y )θ1,0 + z (θ1, 1 + θ1, i ) + (x − y )θ1, 1+i 2 2 2 2 2 2 1 1 1 (8) + 22 22 H 22 H H 22 22 H φ12,1 = (x + y )θ1,0 + z (θ1, 1 + θ1, i ) + (x − y )θ1, i+1 2 2 2 2 2 2
+ 1 30 30 H 1 30 H H 1 30 30 H φ16,1 = (x + y )θ1,0 + z (θ1, 1 + θ1, i ) + (x − y )θ1, 1+i 2 2 2 2 2 2
+, cusp 1 6 6 6 H 6 6 6 H φ10,1 = x y z θ1, 1 − x y z θ1, i . 64 2 2 In Remark 4.11 I will also give the initial Fourier series expansions of these Hermitian Jacobi forms.
+ + + +, cusp Note that [25] uses the Hermitian Jacobi forms φ4,1, φ8,1, φ12,1 and φ10,1 to de- + +, cusp +, cusp termine the structure of Jk,1(O) and the cusp forms ψk,1 ∈ Jk,1 (O) for k = 8, 12, 16 +, cusp +, cusp +, cusp to determine the structure of Jk,1 (O). The cusp forms ψk,1 ∈ Jk,1 (O) for k = 8, 12, 16 are defined by
+, cusp + + ψ8,1 := E4φ4,1 − φ8,1
+, cusp + + ψ12,1 := E4φ8,1 − φ12,1
+, cusp + + ψ16,1 := E4φ12,1 − φ16,1,
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 φ6,1 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 φ6,1 using a different
20 approach. I used SAGE [28] and SAGE code written by Martin Raum to calculate several
− − Fourier series coefficients of φ6,1. This allowed us to guess the theta decomposition of φ6,1, − which I then verified directly. The Hermitian Jacobi form φ6,1 will play an important role in the next section.
Lemma 4.10. Let x, y, and z be again the classical theta constants. Then
− H H H H − φ6,1 := h0θ1,0 + h 1 θ 1 + h i θ i + h 1+i θ 1+i ∈ J6,1(O), 2 1, 2 2 1, 2 2 1, 2
where
1 h := − (x2 + y2)(x8 − x6y2 − x4y4 − x2y6 + y8), 0 2
1 6 4 4 h 1 := z (z − 2x ), 2 2
1 6 4 4 h i := z (z − 2x ), 2 2
1 2 2 8 6 2 4 4 2 6 8 h 1+i := − (x − y )(x + x y − x y + x y + y ). 2 2
+ 15 + 2 + 9 + + + Proof. Consider ψ12,1 := 2 E4φ8,1 − 2E4 φ4,1 − 2 φ12,1. Then ψ12,1 ∈ J12,1(O), and let ˆ H ˆ H ˆ H ˆ H + h0θ1,0 + h 1 θ 1 + h i θ i + h 1+i θ 1+i be the theta decomposition of ψ12,1. Recall the following 2 1, 2 2 1, 2 2 1, 2 identities from Chapter 2:
4 4 4 1 8 8 8 1 4 4 4 4 4 4 x = y + z ,E4 = 2 (x + y + z ),E6 = 2 (x + y )(x + z )(y − z ). These identities allow us (with the help of Mathematica) to verify that ˆ h0 = E6h0 ˆ h 1 = E6h 1 2 2 ˆ h i = E6h i 2 2 ˆ h 1+i = E6h 1+i . 2 2 + − Hence ψ12,1 = E6φ6,1. 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 φ6,1 ∈ − J6,1(O).
I end this section with the initial Fourier series expansions of the Hermitian Jacobi
+ − + +,cusp forms φ4,1, φ6,1, φ8,1, and φ10,1 . See also the Appendix for more coefficients of these forms.
21 Remark 4.11. I have the following initial Fourier series expansions:
1 1 1 1 i i i i + 2 0 2 − 2 0 − 2 − 2 0 2 2 0 − 2 φ4,1 = 1 + q 60 + 32 ζ (ζ ) + ζ (ζ ) + ζ (ζ ) + ζ (ζ )
+ ζζ0 + ζ−1(ζ0)−1 + ζ−i(ζ0)i + ζi(ζ0)−i
1+i 0 1−i −1+i 0 −1−i 1−i 0 1+i −1−i 0 −1+i + 12 ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ···
1 1 1 1 i i i i − 2 0 2 − 2 0 − 2 − 2 0 2 2 0 − 2 φ6,1 = 1 + q − 204 − 64 ζ (ζ ) + ζ (ζ ) + ζ (ζ ) + ζ (ζ )
+ ζζ0 + ζ−1(ζ0)−1 + ζ−i(ζ0)i + ζi(ζ0)−i
1+i 0 1−i −1+i 0 −1−i 1−i 0 1+i −1−i 0 −1+i − 12 ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ···
+ 0 −1 0 −1 −i 0 i i 0 −i φ8,1 = 1 + q 364 + ζζ + ζ (ζ ) + ζ (ζ ) + ζ (ζ )
1+i 0 1−i −1+i 0 −1−i 1−i 0 1+i −1−i 0 −1+i + 28 ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ···
+,cusp 1 1 1 1 i i i i 2 0 2 − 2 0 − 2 − 2 0 2 2 0 − 2 φ10,1 = q ζ (ζ ) + ζ (ζ ) − ζ (ζ ) − ζ (ζ )
2 1 0 1 − 1 0 − 1 − i 0 i i 0 − i +q − 18 ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 − ζ 2 (ζ ) 2 − ζ 2 (ζ ) 2
1+2i 0 1−2i −1+2i 0 −1−2i 1−2i 0 1+2i −1−2i 0 −1+2i + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2 + ζ 2 (ζ ) 2
2+i 0 2−i −2+i 0 −2−i 2−i 0 2+i −2−i 0 −2+i − ζ 2 (ζ ) 2 − ζ 2 (ζ ) 2 − ζ 2 (ζ ) 2 − ζ 2 (ζ ) 2 + ···
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
+ − + +,cusp ring of modular forms, and a set of generators is given by the forms φ4,1, φ6,1, φ8,1, φ10,1 . 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 δ δ Consider the Taylor series expansion of a Hermitian Jacobi form φ ∈ Jk,1(O) around (z, w) = (0, 0):
∞ δ X δ µ ν φ (τ, z, w) = χµ,ν(τ)z w . µ,ν=0
a b The modular transformation law (see (i) of Definition 4.3) with M = ( c d ) implies that
aτ + b χδ µ,ν cτ + d 2πic 1 2πic 2 = σ(ε)εk−µ+ν(cτ + d)k+µ+ν χδ (τ) + χδ (τ) + χδ (τ) + ... . µ,ν cτ + d µ−1,ν−1 2! cτ + d µ−2,ν−2
δ Observe that χµ,ν = 0 unless µ − ν is even. The first several coefficients have the following behavior under modular transformations:
aτ + b χδ = σ(ε)εk(cτ + d)kχδ (τ), 0,0 cτ + d 0,0 aτ + b χδ = σ(ε)εk(cτ + d)k+2χδ (τ) + 2πic(cτ + d)k+1χδ (τ) , 1,1 cτ + d 1,1 0,0 aτ + b χδ = σ(ε)εk+2(cτ + d)k+2χδ (τ), 0,2 cτ + d 0,2 aτ + b χδ = σ(ε)εk+2(cτ + d)k+2χδ (τ), 2,0 cτ + d 2,0 aτ + b χδ = σ(ε)εk+4(cτ + d)k+4χδ (τ), 0,4 cτ + d 0,4 aτ + b χδ = σ(ε)εk (cτ + d)k+4χδ (τ) + 2πic(cτ + d)k+3χδ (τ) 2,2 cτ + d 2,2 1,1 1 + (2πic)2(cτ + d)k+2χδ (τ) . 2! 0,0
The following proposition is an immidiate consequence of the above equations.
δ P∞ δ µ ν δ Proposition 4.12. Let φ (τ, z, w) = µ,ν=0 χµ,ν(τ)z w ∈ Jk,1(O).
+ + If k ≡ 0 (mod 4) and δ = +, then χ0,2(τ) = χ2,0(τ) = 0.
− − − − − If k ≡ 0 (mod 4) and δ = −, then χ0,0(τ) = χ1,1(τ) = χ4,0(τ) = χ0,4(τ) = χ2,2(τ) = 0.
− − If k ≡ 2 (mod 4) and δ = −, then χ0,2(τ) = χ2,0(τ) = 0.
+ + + + + If k ≡ 2 (mod 4) and δ = +, then χ0,0(τ) = χ1,1(τ) = χ4,0(τ) = χ0,4(τ) = χ2,2(τ) = 0.
23 Observe that the Fourier series expansion of a Hermitian Jacobi form
∞ δ X X n r 0 r δ φ (τ, z, w) = c(n, r)q ζ (ζ ) ∈ Jk,1(O) n=0 r∈O# nm−|r|2≥0
δ implies that its Taylor series coefficients χµ,ν have a q-expansion of a holomorphic modular form: ∞ (πi)µ+ν X χδ (τ) = rµrνc(n, r) qn. µ,ν µ!ν! r∈O# n−|r|2≥0
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
δ χ(0,0)(τ) ∈ Mk, δ δ χ(2,0)(τ), χ(0,2)(τ) ∈ Sk+2, δ δ χ(4,0)(τ), χ(0,4)(τ) ∈ Sk+4, δ δ 2πi δ 0 ζ1.1(τ) := χ1,1(τ) − k (χ (0,0)) (τ) ∈ Sk+2, δ δ 2πi δ 0 (2πi)2 δ 00 ζ2.2(τ) := χ2,2(τ) − k+2 (χ (1,1)) (τ) + 2(k+1)(k+2) (χ (0,0)) (τ) ∈ Sk+4. Recall the definition of the Jacobi theta function in (2) of Chapter 2:
X πi(a+n)2τ+2πi(n+a)(z+b) θa,b(τ, z) = e . n∈Z
Note that θa,b(2τ, z + w) satisfies the heat equation, i.e.,
∂2 ∂ θ (2τ, z + w) = 2πi θ (2τ, z + w). ∂z2 a,b ∂τ a,b
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
2πi d (2πi)2 d2 θ (2τ) + θ (2τ)(z + w)2 + θ (2τ)(z + w)4 + ... a,0 2! dτ a,0 4! dτ 2 a,0
and θb,0(2τ, i(w − z)) has a Taylor series expansion of the form
2πi d (2πi)2 d2 θ (2τ) − θ (2τ)(w − z)2 + θ (2τ)(w − z)4 + ... b,0 2! dτ b,0 4! dτ 2 b,0
24 For convenience I write d d2 T := θ (2τ),T 0 := 2πi θ (2τ),T 00 := (2πi)2 θ (2τ), 2a a,0 2a dτ a,0 2a dτ 2 a,0
0 00 and T2b, T2b , and T2b are defined analogously. Then I have
0 0 θa,0(2τ, z + w)θb,0(2τ, i(w − z)) = T2aT2b + (T2aT2b + T2a T2b)zw 1 + (T 0T − T T 0)(z2 + w2) 2 2a 2b 2a 2b 1 + (T 00T + T T 00 + 2T 0T 0)z2w2 4 2a 2b 2a 2b 2a 2b 1 + (T 00T + T T 00 − 6T 0T 0)(z4 + w4) + ··· 4! 2a 2b 2a 2b 2a 2b Furthermore, one easily verifies the following factorization of theta functions:
H a (9) θ ,0(2τ, z + w)θ b ,0(2τ, i(w − z)) = θ a+bi (τ, z, w). 2 2 1, 2
Now I expand the theta decomposition
δ X H φ (τ, z, w) = h a+bi (τ)θ a+bi (τ, z, w) 2 1, 2 a,b=0,1 X a = h a+bi (τ)θ ,0(2τ, z + w)θ b ,0(2τ, i(w − z)) (9) 2 2 2 a,b=0,1 into a Taylor series, and compare its coefficients with the coefficients of the Taylor series
δ δ δ δ δ 2 2 φ (τ, z, w) = χ0,0(τ) + χ1,1(τ)zw + χ0,2(τ) + χ2,0(τ) (z + w )
δ 2 2 δ δ 4 4 + χ2,2(τ)z w + χ0,4(τ) + χ4,0(τ) (z + w ) + ··· . I obtain
δ δ δ δ δ δ δ 0 (χ , χ , χ + χ , χ , χ + χ ) = (h0, h 1 , h i , h 1+i )A , 0,0 1,1 2,0 0,2 2,2 4,0 0,4 2 2 2 where A0 equals the matrix
2 0 1 00 02 1 00 02 T0 2T0T0 0 2 (T0T0 +T0 ) 4! (2T0 T0−6T0 ) 0 0 1 0 0 1 00 00 0 0 1 00 00 0 0 T0T1 T0T1 +T0 T1 2 (T1 T0−T1T0 ) 4 (T1T0 +T0T1 +2T1 T0 ) 4! (T1T0 +T0T1 −6T1 T0 ) 0 0 1 0 0 1 00 00 0 0 1 00 00 0 0 . T0T1 T0T1 +T0 T1 − 2 (T1 T0−T1T0 ) 4 (T1T0 +T0T1 +2T1 T0 ) 4! (T1T0 +T0T1 −6T1 T0 ) 2 0 1 00 02 1 00 02 T1 2T1T1 0 2 (T1 T1+T1 ) 4! (2T1 T1−6T1 ) Equivalently, I find that
δ δ δ 1 δ δ (10) (χ0,0, χ1,1, χ2,0, (χ2,2 − 12χ4,0)) = (h0, h 1 , h i , h 1+i )A, 2 2 2 2
25 where 2 0 02 T0 2T0T0 0 T0 0 0 1 0 0 0 0 T0T1 T0T1 + T0 T1 (T1 T0 − T1T0 ) 2T0 T1 4 (11) A := . T T T T 0 + T 0T − 1 (T 0T − T T 0) 2T 0T 0 0 1 0 1 0 1 4 1 0 1 0 0 1 2 0 02 T1 2T1T1 0 T1
Note that 1 det A = − (T T 0 − T T 0)2((T T 0)2 − 4T T 0T T 0). 2 1 0 0 1 1 0 0 0 1 1 Consider the q-expansions
4 T0 = 1 + 2q + 2q + ···
0 2 2 4 T0 = −8π q − 32π q − · · ·
1 9 T1 = 2q 4 + 2q 4 + ···
1 9 0 2 4 2 4 T1 = −2π q − 18π q − · · · to find that
det A 6= 0.
Now I am in a position to determine the structure of Hermitian Jacobi forms of index
δ δ 1. As before, χµ,ν denote the Taylor coefficients of a Hermitian Jacobi form, and ζµ,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) = Jk,1(O) ⊕ Jk,1(O) → Mk ⊕ Sk+2 ⊕ Sk+2 ⊕ Sk+4
+ + − + + φ 7→ (χ0,0, ζ1,1, χ2,0, ζ2,2 − 12χ4,0)
and
+ − η : Mk−4 ⊕ Mk−6 ⊕ Mk−8 ⊕ Mk−10 → Jk,1(O) = Jk,1(O) ⊕ Jk,1(O)
+ − + +,cusp (e, f, g, h) 7→ (eφ4,1 + fφ6,1 + gφ8,1, hφ10,1 )
are isomorphisms.
26 Assume that k ≡ 2 (mod 4). Then both linear maps
+ − ζ : Jk,1(O) = Jk,1(O) ⊕ Jk,1(O) → Mk ⊕ Sk+2 ⊕ Sk+2 ⊕ Sk+4
− − + − − φ 7→ (χ0,0, ζ1,1, χ2,0, ζ2,2 − 12χ4,0)
and
+ − η : Mk−4 ⊕ Mk−6 ⊕ Mk−8 ⊕ Mk−10 → Jk,1(O) = Jk,1(O) ⊕ Jk,1(O)
+,cusp + − + (e, f, g, h) 7→ (hφ10,1 , eφ4,1 + fφ6,1 + gφ8,1)
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 χ0,0 = ζ1,1 = χ2,0 = + + ζ2,2 − 12χ4,0 = 0. Then Proposition 4.13 implies that
1 χ+ = 0, ζ+ = χ+ , and hence (χ+ − 12χ+ ) = 0 . 1,1 2,2 2,2 2 2,2 4,0
+ − − − − Moreover, Proposition 4.12 gives that χ2,0 = 0 and χ0,0 = χ1,1 = χ2,2 = χ4,0 = 0. Thus, for δ = ± I find that
δ δ δ 1 δ δ (0, 0, 0, 0) = (χ0,0, χ1,1, χ2,0, (χ2,2 − 12χ4,0)) = (h0, h 1 , h i , h 1+i )A, 2 (10): 2 2 2
where A as in (11). Recall that det A 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
+ − + +,cusp + suppose that eφ4,1 + fφ6,1 + gφ8,1 + hφ10,1 = 0. Observe the theta decompositions of φ4,1, + +,cusp − φ8,1, and φ10,1 in (8) and of φ6,1 in Lemma 4.10 to find that
(e, f, g, h)H = (0, 0, 0, 0),
27 where 1 (x6 + y6) 1 z6 1 z6 1 (x6 − y6) 2 2 2 2 h0 h 1 h i h 1+i 2 2 2 H := 1 (x14 + y14) 1 z14 1 z14 1 (x14 − y14) 2 2 2 2 1 6 6 6 1 6 6 6 0 64 x y z − 64 x y z 0
with h0, h 1 , h i , and h 1+i as in Lemma 4.10. With the help of Mathematica (observing the 2 2 2 identity x4 = y4 + z4) one finds that 9 det H = − x16y16z16. 128 Recall the q-expansions of the theta constants (3). Specifically,
1 x = 1 + 2q 2 + ···
1 y = 1 − 2q 2 + ···
1 9 z = 2q 8 + 2q 8 + ··· and I find that x16y16z16 6= 0 . Hence det H 6= 0, which shows e = f = g = h = 0. Finally, Theorem 2.4 implies that
dim Mk + dim Sk+2 + dim Sk+2 + dim Sk+4 = dim Mk−4 + dim Mk−6 + dim Mk−8 + dim Mk−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 Jk,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])
1 ∂ ∂2 (12) L := − 2πim − . m π2 ∂τ ∂w∂z
The following lemma gives its action on Hermitian Jacobi forms.
δ Lemma 5.1. If φ ∈ Jk,m(O), then
(k − 1)m L (φ) = E φ + φˆ m 3 2
ˆ −δ where E2 is the quasimodular Eisenstein series, and where φ ∈ Jk+2,m(O).
˜ (k−1)m Remark 5.2. Observe that the “corrected” heat operator Lm := Lm − 3 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) ˆ (k−1)m of Definition 4.3 with parity δ, then φ := Lm(φ) − 3 E2φ satisfies the modular property a b aτ+b z −1w with parity −δ. Let ε ( c d ) ∈ Γ(O) and set η := cτ+d , cτ+d , cτ+d . I have
2 2 2 ∂ φ 2 ∂ 2 ∂ k k 2πimczw (η) = (cτ + d) φ(η) = (cτ + d) σ() (cτ + d) e cτ+d φ ∂w∂z ∂w∂z ∂w∂z
k k+2 ∂ ∂φ 2πimcw 2πimczw = σ() (cτ + d) + φ e cτ+d ∂w ∂z cτ + d 2 2 2 2 k k+2 2πimczw ∂ φ 2πimc ∂φ ∂φ 4π m c zw = σ() (cτ + d) e cτ+d + φ + w + z − φ . ∂w∂z cτ + d ∂w ∂z (cτ + d)2 Furthermore,
2 ∂ k 2πimczw k−1 k 2πimc zw k ∂φ φ(η) = σ() e cτ+d (cτ + d) kcφ − (cτ + d) φ + (cτ + d) ∂τ (cτ + d)2 ∂τ
∂φ k−1 k+1 2πicmzw ∂φ 2πimcw (13) (η) = σ() (cτ + d) e cτ+d + φ ∂z ∂z cτ + d
∂φ k+1 k+1 2πimczw ∂φ 2πimcz (η) = σ() (cτ + d) e cτ+d + φ ∂w ∂w cτ + d and
∂φ ∂ ∂φ ∂φ (14) (η) = (cτ + d)2 φ(η) + cz (η) + −1cw (η). ∂τ ∂τ ∂z ∂w
Substituting (13) in (14) yields ∂φ k k 2πimczw 2 2 ∂φ (η) = σ() (cτ + d) e cτ+d (cτ + d)kcφ + 2πimc zwφ + (cτ + d) ∂τ ∂τ ∂φ ∂φ + (cτ + d) cz + cw . ∂z ∂w I find that
(k − 1)m aτ + b φˆ(η) = L φ(η) − E φ(η) m 3 2 cτ + d −2mi ∂φ 1 ∂2φ (k − 1)m aτ + b = (η) + (η) − E φ(η). π ∂τ π2 ∂w∂z 3 2 cτ + d
∂φ ∂2φ Substituting ∂τ (η), ∂w∂z (η), and using (1) of Chapter 2 gives
−2 k+2 k+2 2πimczw φˆ(η) = σ() (cτ + d) e cτ+d φ,ˆ
i.e., φˆ 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 φˆ. 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
∂ ∂φ ∂τ ∂φ ∂ ∂φ ∂ φ(γ) = (γ) + (γ) (z + λτ + µ) + (γ) (w + λτ + µ) ∂τ ∂τ ∂τ ∂z ∂τ ∂w ∂τ ∂φ ∂φ ∂φ = (γ) + λ (γ) + λ (γ), ∂τ ∂z ∂w
i.e.,
∂φ ∂ ∂φ ∂φ (16) (γ) = φ(γ) − λ (γ) − λ (γ). ∂τ ∂τ ∂z ∂w
On the other hand, by (15)
∂ ∂φ φ(γ) = − 2πimλλφ e−2πimΛ ∂τ ∂τ ∂ ∂φ (17) φ(γ) = − 2πimλφ e−2πimΛ ∂z ∂z ∂ ∂φ φ(γ) = − 2πimλφ e−2πimΛ. ∂w ∂w
Furthermore,
∂φ ∂ (γ) = φ(γ) ∂z ∂z ∂φ ∂ (18) (γ) = φ(γ) ∂w ∂w ∂2φ ∂2 (γ) = φ(γ). ∂w∂z ∂w∂z
Equations (16), (17), and (18) imply that
∂φ ∂φ ∂φ ∂φ (γ) = + 2πimλλφ − λ − λ e−2πimΛ ∂τ ∂τ ∂z ∂w (19) ∂2φ ∂2φ ∂φ ∂φ (γ) = − 2πimλ − 2πimλ − 4π2m2λλφ e−2πimΛ, ∂w∂z ∂w∂z ∂w ∂z
31 and I find that −1 ∂φ ∂2φ L φ(γ) = 2πim (γ) − (γ) m π2 ∂τ ∂w∂z −1 ∂φ ∂2φ = e−2πimΛ 2πim − = e−2πimΛL (φ). π2 ∂τ ∂w∂z m Hence (k − 1)m φˆ(γ) = L φ(γ) − E (τ)φ(γ) = e−2πimΛφ,ˆ m 3 2 i.e., φˆ satisfies the elliptic property. I conclude that φˆ 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(φ4,1) = E2φ4,1 − φ6,1 5 8 L(φ− ) = E φ− − E φ+ + φ+ 6,1 3 2 6,1 3 4 4,1 8,1 (20) 7 14 7 L(φ+ ) = E φ+ − E φ+ − E φ− 8,1 3 2 8,1 9 6 4,1 9 4 6,1 +,cusp +,cusp L(φ10,1 ) = 3E2φ10,1
In the following, let m = 1, and for convenience, I write L := L1. I denote with Fp the δ field Z/pZ, with Z(p) the ring of p-integral rationals, and with Jk,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) = P c(n, r)qnζr(ζ0)r and ψ(τ, z, w) = P c0(n, r)qnζr(ζ0)r with p-integral rational coefficients, I write φ ≡ ψ (mod p) whenever c(n, r) ≡ c0(n, r) (mod p) for all n, r.
+ − + +,cusp Lemma 5.4. The generators φ4,1, φ6,1, φ8,1, and φ10,1 are linearly independent over Fp.
+ − + +,cusp Proof. Let a, b, c, d ∈ Fp and assume that aφ4,1 +bφ6,1 +cφ8,1 +dφ10,1 ≡ 0 (mod p). Recall + − + +,cusp that Remark 4.11 gives the initial Fourier series expansions of φ4,1, φ6,1, φ8,1, and φ10,1 . In particular, their coefficients of q0ζ0(ζ0)0 are 1, 1, 1, 0, respectively, their coefficients of
1 1 0 1 1 i 0 −i q ζ 2 (ζ ) 2 are 32, −64, 0, 1, respectively, their coefficients of q ζ 2 (ζ ) 2 are 32, −64, 0, −1,
32 1 1−i 0 1+i respectively, and their coefficients of q ζ 2 (ζ ) 2 are 12, −12, 28, 0, respectively. Comparing
1 1 0 1 1 i 0 −i the coefficients of q ζ 2 (ζ ) 2 and q ζ 2 (ζ ) 2 gives the system of equations:
32a − 64b + d ≡ 0 (mod p)
32a − 64b − d ≡ 0 (mod p),
0 0 0 0 1 1−i 0 1+i and I find that d ≡ 0 (mod p). Comparing the coefficients of q ζ (ζ ) , q ζ 2 (ζ ) 2 , and
1 1 0 1 q ζ 2 (ζ ) 2 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)
+ − + +,cusp and since p 6= 2, 3, I find that a ≡ b ≡ c ≡ 0 (mod p), i.e., φ4,1, φ6,1, φ8,1 , and φ10,1 are linearly independent over Fp.
δ + − + +,cusp Proposition 5.5. If φ ∈ Jk,1(Z(p)) such that φ = eφ4,1 + fφ6,1 + gφ8,1 (or φ = hφ10,1 ), 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)).
+ − + +,cusp Proof. Suppose that φ = eφ4,1 + fφ6,1 + gφ8,1 (the case φ = hφ10,1 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
t t + t − t + 0 ≡ p φ ≡ p eφ4,1 + p fφ6,1 + p gφ8,1 (mod p). This yields a nontrivial linear dependence + − + relation for φ4,1, φ6,1, and φ8,1, 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 φ4,1, φ6,1, and φ8,1, 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 m0 are congruent modulo p, then m = m0. The following corollary is analogous to Sofer’s Lemma 2.1 in the case m = 1.
δ δ0 Corollary 5.6. Let φ ∈ Jk,1(Z(p)) and ψ ∈ Jk0,1(Z(p)) such that 0 6≡ φ ≡ ψ (mod p). Then k ≡ k0 (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.
δ δ0 0 Remark 5.7. Let φ ∈ Jk,1(Z(p)) and ψ ∈ Jk0,1(Z(p)) such that φ ≡ ψ (mod p). If δ 6= δ and k ≡ k0 (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.
δ n δ o Definition 5.8. Set Jgk,1 := φ (mod p): φ ∈ Jk,1(Z(p)) . For Hermitian Jacobi forms with p-integral rational coefficients, I define the filtration modulo p by
n δ o Ω(φ) := inf k : φ (mod p) ∈ Jgk,1 .
Next I define the U(p) operator for Hermitian Jacobi forms.
X Definition 5.9. For φ(τ, z, w) = c(n, r)qnζr(ζ0)r, I define: # n∈Z,r∈O nm−|r|2≥0
X n r 0 r φ(τ, z, w) U(p) := c(n, r)q ζ (ζ ) . # n∈Z,r∈O nm−|r|2≥0 p|4(nm−|r|2)
34 P n r 0 r P 2 n r 0 r Observe that if φ = c(n, r)q ζ (ζ ) , then Lm(φ) = 4(nm − |r| )c(n, r)q ζ (ζ ) . Thus, Fermat’s little theorem yields that
p−1 Lm (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.
δ Proposition 5.10. If φ ∈ Jk,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
δ Ep+1 ≡ E2 (mod p). Lemma 5.1 shows that L(φ) (mod p) ∈ J^k+p+1,1 if p ≡ 3 (mod 4) and ^−δ L(φ) (mod p) ∈ Jk+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
k−1 hand, if Ω L(φ) < k + p + 1, then Ω 3 φE2 ≤ 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
+ − + +,cusp φ = eφ4,1 + fφ6,1 + gφ8,1 (or φ = hφ10,1 ),
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 δ Theorem 5.11. Let φ ∈ Jk,1(Z(p)) such that φ 6≡ 0 (mod p). If p > k , then
2p + 4 − k, if φ U(p) 6≡ 0 (mod p), Ω Lp+2−k(φ) = 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
A φ1 a low point of its heat cycle if it occurs directly after a fall, i.e., if φ1 = L (φ) and
A−1 Ω(L (φ)) ≡ 1 (mod p). Let φ1 be a low point of its heat cycle and let cj ∈ N be minimal such that cj −1 Ω L (φ1) = Ω(φ1) + (cj − 1)(p + 1) ≡ 1 (mod p),
and let bj ∈ N be given by cj Ω L (φ1) = Ω(φ1) + cj(p + 1) − bj(p − 1).
P P 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 X X X (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
δ if c1 = B = 1. However, this is impossible if φ1 = φ, since Jk,1 = {0} if k < 4. For the case
36 c1 = p + 2 − B I find that
c1 Ω(L (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
p+2−k c1 ≥ 1 implies k < 1. Therefore, c1 = p + 2 − B, and L (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
+ + φ8,1 − E4φ4,1 + 8 5
+,cusp φ10,1 + 10 5, 23
+ − E6φ4,1 − E4φ6,1 - 10 5, 7
+ 2 + E4φ8,1 − E4 φ4,1 + 12 5
+,cusp E4φ10,1 + 14 5
+,cusp E6φ10,1 - 16 5,11,13
2 +,cusp E4 φ10,1 + 18 5, 7, 13, 23 , 79
37 The proof of the U(p) congruences in Table 5.1 relies on the identities 1 L(E ) = (E2 − E ) 2 3 2 4 4 (21) L(E ) = (E E − E ) 4 3 2 4 6 2 L(E6) = 2(E2E6 − E4 )
and also the identities from Example 5.3:
+ + − L(φ4,1) = E2φ4,1 − φ6,1 5 8 L(φ− ) = E φ− − E φ+ + φ+ 6,1 3 2 6,1 3 4 4,1 8,1 (22) 7 14 7 L(φ+ ) = E φ+ − E φ+ − E φ− 8,1 3 2 8,1 9 6 4,1 9 4 6,1 +,cusp +,cusp L(φ10,1 ) = 3E2φ10,1 .
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,
4 +,cusp 2 2 +,cusp +,cusp L φ10,1 ≡ 2E2 E4 + E2 + 3E2E6 φ10,1 ≡ φ10,1 (mod 5).
+,cusp +,cusp 2 +,cusp +,cusp +,cusp Observe that φ10,1 ≡ E4φ10,1 ≡ E4 φ10,1 (mod 5). Thus, if φ = φ10,1 , φ = E4φ10,1 , 2 +,cusp or φ = E4 φ10,1 , then L4(φ) ≡ φ (mod 5).
Moreover,
4 + + 2 + + + + L (φ8,1 − E4φ4,1) ≡ E4 φ8,1 + 4E4φ4,1 ≡ φ8,1 − E4φ4,1 (mod 5) and hence
4 + 2 + 4 + + + + + 2 + L (E4φ8,1 − E4 φ4,1) ≡ L (φ8,1 − E4φ4,1) ≡ φ8,1 − E4φ4,1 ≡ E4φ8,1 − E4 φ4,1 (mod 5).
Finally,
4 +,cusp +,cusp +,cusp L E6φ10,1 ≡ E4E6φ10,1 ≡ E6φ10,1 (mod 5)
38 and
4 + − + − + − L (E6φ4,1 − E4φ6,1) ≡ E4 E6φ4,1 + 4E4φ6,1 ≡ E6φ4,1 − E4φ6,1 (mod 5) .
Consider p = 7. Note that E6 ≡ 1 (mod 7) and E2 ≡ E8 (mod 7). Direct calcula- tions show that
6 + − 2 + − + − L (E6φ4,1 − E4φ6,1) ≡ E6 E6φ4,1 + 6E4φ6,1 ≡ E6φ4,1 − E4φ6,1 (mod 7)
and
6 2 +,cusp 2 +,cusp L E4 φ10,1 ≡ E4 φ10,1 (mod 7) .
3 2 Consider p = 11. Note that E10 ≡ E4E6 ≡ 1 (mod 11) and E2 ≡ E12 ≡ 5E4 + 7E6 (mod 11). A direct calculation shows that
10 +,cusp 9 5 10 6 2 2 3 L E6φ10,1 ≡ 9E4 + 3E4 E6 + 2E4 E6 + 4E4 E6 + 3E4 E6
7 3 3 4 4 5 6 7 +,cusp + 4E4 E6 + 10E4 E6 + 10E4 E6 + 10E6 + E4E6 φ10,1
9 4 9 4 ≡ (9E4 + 3E4 + 2E4 + 4E4 + 3E6
4 6 6 +,cusp + 4E4 + 10E6 + 10E6 + 10E6 + E6 )φ10,1
9 4 6 +,cusp ≡ (11E4 + 11E4 + 23E6 + 11E6 )φ10,1
+,cusp ≡ E6φ10,1 (mod 11).
3 2 2 Consider p = 13. Note that E12 ≡ 6E4 + 8E6 ≡ 1 (mod 13) and E2 ≡ E14 ≡ E4 E6 (mod 13). Direct calculations show that
12 +,cusp 6 3 2 4 +,cusp 2 +,cusp +,cusp L E6φ10,1 ≡ E6 10E4 + 5E4 E6 + 12E6 φ10,1 ≡ E6E12φ10,1 ≡ E6φ10,1 (mod 13) and
12 2 +,cusp 2 6 3 2 4 9 2 6 4 L E4 φ10,1 ≡ E4 10E4 + 9E4 E6 + 6E6 + 8E4 E6 + 11E4 E6
3 6 6 2 3 4 6 +,cusp + 12E4 E6 + 5E4 E6 + 9E4 E6 + 9E6 φ10,1
2 9 6 2 3 4 6 +,cusp ≡ E4 E12(8E4 + 6E4 E6 + 8E4 E6 + 5E6 )φ10,1
39 2 2 6 3 2 4 +,cusp ≡ E4 E12(10E4 + 5E4 E6 + 12E6 )φ10,1
2 3 3 2 +,cusp ≡ E4 E12(6E4 + 8E6 )φ10,1
2 4 +,cusp ≡ E4 E12φ10,1
2 +,cusp ≡ E4 φ10,1 (mod 13) .
+,cusp Consider p = 23. I apply Theorem 5.11 to verify the U(p) congruences for φ10,1 and 2 +,cusp 4 3 E4 φ10,1 . One finds that E22 ≡ 10E4 E6 + 14E4E6 ≡ 1 (mod 23). +,cusp Let φ = φ10,1 . Observe that p + 2 − k = 15. A direct calculation shows that
15 +,cusp 2 4 3 +,cusp 2 +,cusp 2 +,cusp L φ10,1 ≡ 12E4 10E4 E6 + 14E4E6 φ10,1 ≡ 12E4 E22φ10,1 ≡ 12E4 φ10,1 (mod 23).
23+2−10 +,cusp +,cusp Hence Ω L φ10,1 = 18 = 23 + 5 − 10, and φ10,1 | U(23) ≡ 0 (mod 23).
2 +,cusp Let φ = E4 φ10,1 . Observe that p + 2 − k = 10. A direct calculation shows that
7 2 +,cusp 4 3 +,cusp +,cusp +,cusp L E4 φ10,1 ≡ 20E4 E6 + 5E4E6 φ10,1 ≡ 2E22φ10,1 ≡ 2φ10,1 (mod 23).
23+2−18 2 +,cusp 2 +,cusp Hence Ω L E4 φ10,1 = 10 = 23+5−18, and E4 φ10,1 | U(23) ≡ 0 (mod 23).
2 +,cusp Consider p = 79. I apply Theorem 5.11 to verify the U(p) congruences for E4 φ10,1 . One finds that
18 15 3 12 5 9 7 6 9 3 11 13 E78 ≡ 26E4 E6 +10E4 E6 +73E4 E6 +33E4 E6 +41E4 E6 +72E4 E6 +62E6 ≡ 1 (mod 79).
A direct calculation shows that
63 2 +,cusp 32 29 3 26 5 23 7 20 9 17 11 L E4 φ10,1 ≡ 73E4 E6 + 46E4 E6 + 70E4 E6 + 12E4 E6 + 57E4 E6 + 75E4 E6
14 13 11 15 8 17 5 19 2 21 +,cusp + 61E4 E6 + 9E4 E6 + 16E4 E6 + 39E4 E6 + 31E4 E6 φ10,1
14 11 2 8 4 5 6 2 8 +,cusp ≡ E78 18E4 + 7E4 E6 + 71E4 E6 + 37E4 E6 + 40E4 E6 φ10,1
14 11 2 8 4 5 6 2 8 +,cusp ≡ 18E4 + 7E4 E6 + 71E4 E6 + 37E4 E6 + 40E4 E6 φ10,1 (mod 79).
79+2−18 2 +,cusp 2 +,cusp Hence Ω L E4 φ10,1 = 66 = 79 + 5 − 18, and E4 φ10,1 |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 φ4,1, φ6,1, and φ8,1, 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.
δ P n r 0 Let φ = c(n, r)q ζ ζ ∈ 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).
δ +,cusp If φk,1 is not generated by φ10,1 (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.
δ +,cusp On the other hand, if φ is generated by the cusp form φ10,1 (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<(r) ≡ 0 (mod 2) and 2<(r) ≡ 1 (mod 2) yield coefficients c(n, r) that differ only be a sign. Table A.3 lists the coefficients corresponding to 2<(r) ≡ 1 (mod 2), and I abuse notation and label them as c(D).
42 Table A.1. Fourier coefficients of non-cusp forms
+ − + coeff. φ4,1 φ6,1 φ8,1 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. φ4,1 φ6,1 φ8,1 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. φ4,1 φ6,1 φ8,1 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. φ4,1 φ6,1 φ8,1 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. φ4,1 φ6,1 φ8,1 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. φ4,1 φ6,1 φ8,1 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)
1 + + 1 + − 1 + 2 + coeff. 16 (φ8,1 − E4φ4,1) 24 (E6φ4,1 − E4φ6,1) 16 (E4φ8,1 − E4 φ4,1) 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 1 + + 1 + − 1 + 2 + coeff. 16 (φ8,1 − E4φ4,1) 24 (E6φ4,1 − E4φ6,1) 16 (E4φ8,1 − E4 φ4,1) 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 1 + + 1 + − 1 + 2 + coeff. 16 (φ8,1 − E4φ4,1) 24 (E6φ4,1 − E4φ6,1) 16 (E4φ8,1 − E4 φ4,1) 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 1 + + 1 + − 1 + 2 + coeff. 16 (φ8,1 − E4φ4,1) 24 (E6φ4,1 − E4φ6,1) 16 (E4φ8,1 − E4 φ4,1) 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 1 + + 1 + − 1 + 2 + coeff. 16 (φ8,1 − E4φ4,1) 24 (E6φ4,1 − E4φ6,1) 16 (E4φ8,1 − E4 φ4,1) 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 1 + + 1 + − 1 + 2 + coeff. 16 (φ8,1 − E4φ4,1) 24 (E6φ4,1 − E4φ6,1) 16 (E4φ8,1 − E4 φ4,1) 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)
+,cusp +,cusp +,cusp 2 +,cusp coeff. φ10,1 E4φ10,1 E6φ10,1 E4 φ10,1 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 +,cusp +,cusp +,cusp 2 +,cusp coeff. φ10,1 E4φ10,1 E6φ10,1 E4 φ10,1 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 -47330347275 1111185425925 -12115027894635 c(183) -23826622 35496554258 -58526672422 150205654477538 c(187) 6799950 79751688750 333355398750 -73872767128050 c(191) 7601040 3023600400 -461688433200 378445125090960 c(195) 14132100 1184314500 6392857282500 1112525700 c(199) 7375230 -60914130450 5607134814150 -365457559104930
56 +,cusp +,cusp +,cusp 2 +,cusp coeff. φ10,1 E4φ10,1 E6φ10,1 E4 φ10,1 c(203) -989604 36310537116 -15931528010244 -334024893520164 c(207) -16889958 39439408842 1489644569538 -1060947697158 c(211) -19556595 -156969021075 -8903703802275 358736662903245 c(215) -1794690 -862206930 3868652730150 -76214855970 c(219) 6516610 -5982269150 14644904168050 -900413927701310 c(223) 16045416 106000269096 2695602982536 1368049981267176 c(227) -10484343 7216936137 12397559772393 1354276509585417 c(231) 48988800 299076624000 -30317208768000 -1067144013936000 c(235) -10299960 -5819956920 170791691400 143057490120 c(239) -31037580 -350804544300 28175453756100 -2676206155766220 c(243) 6486723 -124342186077 -19835751730797 -16876949393277 c(247) -21198150 -291801933750 -18178298628750 106168883813850 c(251) -924885 235907581275 -11717522369325 2774022488324235 c(255) -25688700 10633558500 -4842071347500 41489900100 c(259) -14287140 799205339100 65757206598300 -2316845182727460 c(263) 52029486 -122103899874 2929965115446 5363291853041166 c(267) 86795778 -753989286942 3585960554322 -153808569932862 c(271) 45900 285917899500 13793346373500 -507716750128500 c(275) -17620875 494237143125 -41046193546875 -8150482129660875 c(279) -112599720 -37983508200 -37674643422600 28726306504920 c(283) 9017955 -1055943465645 -82293108777045 3110663386988355 c(287) 1360476 471719971836 21634120862604 -3063049036066404 c(291) -46670270 -970603845950 65197469070850 3929332845647170 c(295) 78420150 -8233508250 121917595398750 -674553321450 c(299) -34415820 1292560845300 -29054336021100 -200247283330380
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