Mock-modular forms of weight one - UCLA Department of Mathematics

MOCK-MODULAR FORMS OF WEIGHT ONE 51
We can now easily deduce Theorem 1.2. It is convenient to give a slightly more general
version stated in terms of theta series and quadratic forms.
Corollary 9.3. Let p be a prime number congruent to 3 modulo 4, and −d a discriminant
not of the form −pm 2 for any integer m. For a class B ∈ Cl(F ), suppose ˜ϑ B 2(z) =
∑
n∈Z r+ B
(n)q n ∈ M − 2 1 (p) has shadow ϑ B 2(z) and ord ∞ ( ˜ϑ B 2) > −p/4. Then
∣
(110) log
∣
∏
q∈Q −d /Γ
(j(τ B ) − j(τ q )) 1/wq ∣ ∣∣∣∣∣
= − 1 4
∑
Remark. When −d is fundamental, w d = 2w q for all q ∈ Q −d .
k∈Z
δ(k) r + B 2 ( pd−k2
4
).
Proof. Let f(z) = f d (z) with d not of the form pm 2 . Using the Borcherds lift of f d (z) in (99),
the right hand side of (102) can be rewritten as
∣
−4
∑
∏
∣∣∣∣∣
ψ 2 (A) log
(j(τ A ) − j(τ q )) 1/wq .
∣
A∈Cl(F )
q∈Q −d /Γ
For each non-trivial character ψ, choose ˜g ψ (z) ∈ M − 1 (p) having shadow g ψ (z) and r + ψ (n) = 0
for all n ≤ −p/4. For these ˜g ψ (z), equation (101) becomes
〈f lift,θ
d
, g ψ 〉 reg = ∑ ( ) pd − k
r + 2
ψ
δ(k).
4
k∈Z
Combining these two together and dividing both sides by -4, we have
∣
∑
( ) pd − k
(111) − 1 r + 2
4 ψ
δ(k) =
∑
∣∣∣∣∣ ∏
ψ 2 (A) log
4
(j(A) − j(τ q )) 1/wq ,
∣
k∈Z
A∈Cl(F )
which is precisely equation (15) when −d is fundamental.
Consider the mock-modular form ˜F (z) ∈ M − 1 (p) defined by
⎛
˜F (z) := 1
H(p)
⎜
⎝ E+ 1 (z) +
∑
ψ:Cl(F )→C ×
ψ non-trivial
q∈Q −d /Γ
⎞
ψ(B −2 )˜g ψ (z) ⎟
⎠ − ˜ϑ B 2(z).
∑
Since ϑ B 2(z) = 1
H(p) ψ ψ(B−2 )g ψ (z), ˜F (z) has shadow F (z) = 0, hence is modular. Also, it
has Fourier expansion in the form
˜F (z) =
∑
c + (n)q n .
From the proof of Lemma 9.2, we know that
∑ (
c +
k∈Z
pd−k 2
4
n>−p/4
χ p(n)≠1
)
δ(k) = 〈f lift,θ
d
, F 〉 reg = 0.