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

46 W. DUKE AND Y. LI
be a mock-modular form with shadow g ψ (z). By Proposition 3.4, the regularized inner
product 〈f lift,θ , g ψ 〉 reg can be expressed in terms of r + ψ
(n) as
(101) 〈f lift,θ , g ψ 〉 reg = ∑ ∑
( )
−pm − k
r + 2
ψ
c(f, m)δ(k).
4
m∈Z
k∈Z
The following theorem relates this inner product, hence the linear combination of r + ψ
(n), to
the values of the Borcherds lift Ψ f (z).
Theorem 9.1. Let f(z) ∈ M
1/2 ! be a weakly holomorphic modular form with integral Fourier
coefficients c(f, n), and Ψ f (z) its Borcherds lift. Suppose ψ is a non-trivial character of
Cl(F ). Then we have
∑
(102) 〈f lift,θ , g ψ 〉 reg = −2 lim ψ 2 (A) ( log |Ψ f (τ A + ɛ)| 2 + C f log |y A | ) ,
ɛ→0
A∈Cl(F )
where C f = ∑ k∈Z c(f, −pk2 ) is the constant term of f lift,θ (z), and τ A is a CM point associated
to the class A.
Remark. Recall that jm
lift,B (z) = j m (pz)ϑ B (z) for B ∈ Cl(F ). When B = A 0 is the principal
class in Cl(F ), we can write
2j lift,A 0
m (z) = j m (pz)U 4 (θ(pz)θ(z))
= (j m (4z)θ(z)) lift,θ
= ∑ t∈Z
f lift,θ
4m−t
(z). 2
t 2 ≤4m
Suppose r A0 (m) = 0, then m is not a perfect square and the right hand side of equation (53)
becomes
−2
∑
ψ 2 (A) log |Ψ m (τ A , τ A )| .
A∈Cl(F )
By Kronecker’s identity, this is the same as
⎛
−2
∑
A∈Cl(F )
∑
ψ 2 (A) ⎜
⎝ log |Ψ f4m−t 2
(τ A )| ⎟
⎠ .
t∈Z
t 2 ≤4m
Thus, this case of Theorem 5.1 is a consequence of Theorem 9.1.
The plan of the proof goes as follows. First, notice that both sides of equation (102) are
additive with respect to f(z). So it suffices to prove the theorem when f(z) = f d (z) for all
d ≥ 0 and d ≡ 3 or 0 modulo 4. For convenience, we denote
(103) a(d, ψ) := 〈f lift,θ
d
, g ψ 〉 reg .
Then, we define a function Φ(z) analogous to G D (z) in [29], where the inner product between
G D (z) and a weight k+1/2 level 4 eigenform g(z) gives special values of L-function associated
to the Shimura lift of g(z). For us, Φ(z) is non-holomorphic, transforms with weight 3/2 and
level 4, and satisfies Kohnen’s plus space condition. Since there is no holomorphic cusp form
of weight 3/2, level 4 satisfying the plus space condition, the inner product between the
holomorphic projection of (almost) Φ(z) and the Poincaré series with an s-variable giving
rise to such cusp forms is zero as s approaches zero. This gives us the identity relating finite
⎞