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

36 W. DUKE AND Y. LI
where x j (A ′ , B) is the preimage of the Heegner point τ j (A ′ , B) on X 0 (N) and
⎛
⎞
∑
T f := ⎜
⎝ c(f, −m)T m − (N + 1) ∑ c(f, −Nm ′ )T m ′W N
⎟
⎠ ,
m≥1
m ′ ≥1
N∤m
N∤m ′ (
U f,B,ψ = c f,1ψ(B[n]) + c f,2 ψ(B −1 [n]) + c f,3 ψ(B[n] −1 ) + c f,4 ψ(B −1 [n] −1 ) ∑
ψ 2 (A ′ ) log |u
12H(p)(N 2 A ′|)
− 1)
A ′
for some constants c f,j ∈ Q determined by f(z). Since f(z) ∈ S 1 , the coefficient c(f, −m)
vanishes for all m ≥ 1 satisfying N 2 |m. So we have N ∤ m ′ in the summation above. Using
(49) and equation (2.16) in [24, p. 240], one can show that the c f,j ’s are integral linear
combinations of c(f, −m), m ≥ 1. Since f is in the integral span of S 1 , Lemma 6.4 implies
that c(f, −m) ∈ Z for all m ≥ 1. Then the denominator of constant in the definition of U f,B,ψ
is independent of f.
Given any newform h(z) ∈ S 2 (N), we know that (h|W N )(z) = −(h|U N )(z). It is then
easy to check that T f (h(z)) = 0. Since N is prime, S 2 (N) is spanned by newforms. So T f
annihilates any h(z) ∈ S 2 (N). Then T f ((x 1 (A, B)) − (0)) is a trivial divisor on J 0 (N) defined
over H, since the actions of the Hecke operators and Fricke involution on the Jacobian are
the same as those on S 2 (N). So it is the divisor of a rational function φ f,A,B on X 0 (N),
defined over H. By the axioms defining height pairings, we can rewrite Σ f,N,B,ψ as
Σ f,N,B,ψ =
∑
ψ 2 (A) log |u f,B (A)| + U f,B,ψ ,
where
A∈Cl(F )
u f,B (A) = φ f,A,B(τ 2 (f, A, B))
∈ H.
φ f,A,B (∞)
Given any σ C ∈ Gal(H/F ) associated to C ∈ Cl(F ), it sends the Heegner point x 1 (A, B) to
the Heegner point x 1 (AC −1 , B). Since the Galois action commutes with the Hecke action, we
see that
φ σ C
f,A,B (z)
φ f,AC −1 ,B(z) ∈ H×
is a constant. So σ C (u f,B (A)) = u f,B (AC −1 ). Thus, the proposition holds for all f(z) in the
integral span S 1 .
□
7. Proof of Theorem 1.1
From §5 and §6, we know that the regularized inner product between f lift,N,B (z) ∈ M !,+
1 (p)
and g ψ (z) can be put into a form quite similar to (10) and satisfies condition (iii) in Theorem
1.1, whenever f(z) ∈ M 0(N) ! has rational Fourier coefficients and N is either 1 or a prime
number different from p. For a mock-modular form ˜g ψ (z) = ∑ n∈Z r+ ψ (n)qn as in Proposition
4.2, (76) gives rise to the following equation involving a linear combination of r + ψ (n)’s
∑
δ(k)r B (pm − Nk)r + ψ (k) =〈f lift,N,B , g ψ 〉 reg
(86)
m∈Z
δ N (m)c(f, −m) ∑ k∈Z
+ N ∑ m ′ ≥0
c(f, −Nm ′ )ρ N,B,ψ (m ′ ).