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

30 W. DUKE AND Y. LI
Let A ∈ Pic(O F ) = Cl(F ). A point τ ∈ H corresponding to the Heegner point (O, n, A)
must satisfy an equation
Aτ 2 + Bτ + C = 0,
with B 2 − 4AC = −p, N|A, B ≡ b n (mod 2N) and gcd(A/N, B, NC) = 1. We will denote
this image by τ(A, n). Notice τ(A, n) is well-defined up to the action of Γ 0 (N).
Let J 0 (N) be the Jacobian of X 0 (N). Its complex points is a compact Riemann surface
and can be viewed as the set of divisors of degree zero modulo the set of principal divisors
on X 0 (N). Let 〈, 〉 C be the height symbol on X 0 (N)(C). It is a unique real-valued function
defined on the set of divisors of degree zero and satisfies
〈∑
ni (x i ), b〉
= ∑ n i log |f(x i )| 2
C
i
if b = (f) is a principal divisor.
For n ≥ 0 and a class B ∈ Cl(F ), denote the inner product
(72) ρ N,B,ψ (n) := 〈j lift,B[n]
n
+ j lift,B[n]−1
n , g ψ 〉 reg .
In particular for j 0 (z) = 1, the inner product ρ N,B,ψ (0) becomes 〈T N (ϑ B ), g ψ 〉, where T N is
the N th Hecke operator on M ! 1(p). Also, notice that
(73) T N (ϑ B )(z) = ϑ B (Nz) + (ϑ B |U N )(z) = ϑ B[n] (z) + ϑ B[n] −1(z).
By Corollary 5.2, we know that ρ N,B,ψ (n) can be written in the form of the right hand
side of (68) and satisfies the conditions in Proposition 6.1. The following lemma relates
〈f lift,N,B , g ψ 〉 reg to ρ N,B,ψ (n) and an infinite sum analogous to the right hand side of (62).
Lemma 6.6. Suppose f(z) ∈ M !,new
0 (N). In the notations above, we have
(74) 〈f lift,N,B , g ψ 〉 reg = ∑ m ′ ≥1
c(f, −Nm ′ )ρ N,B,ψ (m ′ ) + C f ρ N,B,ψ (0) − Σ f,N,B,ψ ,
where
∑
Σ f,N,B,ψ := − lim
s→1
C f =
m≥1
N∤m
(N + 1) lim
(
c(f, −m) ∑ k≥1
s→1
∑
m ′ ≥1
c(f, −Nm ′ ) ∑ k ′ ≥1
Nc(f, 0) + 24 ∑ m ′ ≥1
c(f, −Nm ′ )σ 1 (m ′ )
)
δ(k)r B (pm + Nk)r ψ (k)2Q s−1
(1 + 2kN +
pm
)
δ(k ′ )r B (k ′ )r ψ (Nk ′ − pm ′ )2Q s−1
(−1 + 2k′ N
,
pm ′
)
.
Proof. Let ˜g ψ (z) ∈ M − 1 (p) be a mock-modular form as in Proposition 4.2 with Fourier expansion
˜g ψ (z) = ∑ n∈Z r+ ψ (n)qn and ĝ ψ (z) the associated harmonic Maass form. By Theorem
5.1, we have the following identities
)
ρ N,B,ψ (n) = −24σ 1 (n)ρ N,B,ψ (0) + lim δ(k)c(T N (ϑ B ), pm + k)r ψ (k)2Q s−1
(1 + 2k ,
pm
ρ N,B,ψ (n) = ∑ k∈Z
s→1
∑
k≥1
r + ψ (k) ( (
r pn−k
)
B N + rB (pNn − Nk) ) δ(k).