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

28 W. DUKE AND Y. LI
The proof of Proposition 6.1 follows the same steps as in the proof of Theorem 5.1 and
Corollary 5.2. First, we will express 〈f lift,N,B , g ψ 〉 reg in terms of other regularized inner products
and the limit of an infinite sum. Then, via Green’s function, we will relate the limit of
the infinite sum to the height pairings between Heegner divisors on J 0 (N), the Jacobian of
X 0 (N). Since the Heegner divisor is trivial on J 0 (N), the height pairing is the same as the
value of some modular function evaluated at a Heegner point. From there, we can deduce
equation (68). Before presenting the proof, we need some notations and facts about level N
modular forms. For this part, we only require N to be a prime number.
Let M 2(N), ! resp. M 0(N), ! be the space of weakly holomorphic weight 2 modular forms,
resp. modular functions, of level N, and trivial nebentypus. Denote the subspaces of M 2(N)
!
of cusp forms and holomorphic modular forms by M 2 (N), S 2 (N) respectively. Let M !,new
0 (N)
be the subspace of M 0(N) ! containing weakly holomorphic modular forms f(z) satisfying
(69) (f|W N )(z) = −N(f|U N )(z).
The following lemma shows that M ! 0(N) can be be decomposed into the direct sum of
M !,new
0 (N) and level one modular functions.
Lemma 6.2. Let f(z) be a modular function of prime level N. Then it can be written
uniquely as
f(z) = f 1 (Nz) + f 2 (z),
such that f 1 (z) is a modular function of level 1 and f 2 (z) ∈ M !,new
0 (N).
Proof. Let a, b ∈ Z. With some matrix calculations, we have
( ( ))
f| UN W N + a N (z) =
a−1
f(z) + N TrN 1 (f)(Nz)
( ( ) ( ))
f| UN W N + a UN W
N
N + b N (z) =
(a−1)(b−1)
f(z) + a+b+N−1 Tr N N 2
N 1 (f)(Nz)
Set f j (z) as follows and we are done
f 1 (z) = N
N+1 TrN 1 (f)(z),
f 2 (z) = − N N+1 NW N ) (z) − f(z)) .
Alternatively, one can characterize the space M ! 0(N) via the space S 2 (N).
Definition 6.3. A set of numbers {λ m : m ≥ 1} is called a relation for S 2 (N) if
(i) λ m ∈ C is non-zero for all but finitely many m.
(ii) For any cusp form h(z) = ∑ m≥1 c(h, m)qm ∈ S 2 (N), the numbers {λ m } satisfy
∑
(70)
δ N (m)λ m c(h, m) = 0.
m≥1
where δ N (m) = N + 1 if N|m and 1 otherwise.
A relation {λ m } m≥1 is called integral if λ m ∈ Z for all m ≥ 1. Denote Λ N the set of all
relations for S 2 (N).
For any f(z) ∈ M !,new
0 (N), one knows that the principal part coefficients, {c(f, −m) :
m ≥ 1}, is a relation for S 2 (N) from works by Petersson. By considering the Fourier expansion
of vector-valued Poincaré series, (See [7], [27], [32]), one can also show that the converse
is true. So given λ = {λ m } ∈ Λ N , the expression
∑
λ m q −m
m≥1
□