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

MOCK-MODULAR FORMS OF WEIGHT ONE 33
Proof of Proposition 6.1. If f(z) = f 1 (Nz) with f 1 (z) a modular function of level one, then
(f|W N )(z) = (f|U N )(z) and
f lift,N,B (z) = f lift,B
1 (z) = f 1 (pz)ϑ B (z).
This case of the proposition follows from Corollary 5.2. By Lemma 6.2, it is enough to prove
the proposition for f(z) ∈ M !,new
0 (N). In that case, some calculations show that
(81) f lift,N,B (z) = −N(f|U N )(pz)ϑ B (Nz) + (f(pz)ϑ B (z)) |U N .
Let χ p (N) = ɛ. It is easy to see that f lift,N,B (z) ∈ M !,ɛ
1 (p). Since g ψ (z) ∈ S + 1 (p), Proposition
3.4 tells us that 〈f lift,N,B , g ψ 〉 reg = 0 when ɛ = −1. So we can suppose χ p (N) = ɛ = 1.
By Lemma 6.4, it is easy to see that M !,new
0 (N) is generated by the set S 1 ∪ S 2 , where
(82)
S 1 := {f m (z) : m ≥ 1 + g N , N 2 ∤ m}
S 2 := {f N 2 m(z) − N+1
δ N (m) f m(z) : m ≥ 1 + g N }.
Let f(z) = f N 2 m(z) − N+1
δ N (m) f m(z) ∈ S 2 . We can uniquely write the integer m as N r m ′ with
r ≥ 0, m ′ ≥ −1 and gcd(N, m ′ ) = 1. Then at the cusp infinity, we have
f(z) = q −N r+2 m ′ − N+1
δ N (N r m ′ ) q−N r m ′ + O(1).
From equation (81), it follows that the −n th Fourier coefficient of f lift,N,B (z) can be written
as
(
( ))
( )
pN r m ′
c(f lift,N,B pN
, −n) = −N r r+1 m ′ −n
B − N+1 r N
−n
N
δ N (N r m ′ ) B N
(
)
+ r B (−Nn + pN r+2 m ′ ) − N+1 r δ N (N r m ′ ) B(−Nn + pN r m ′ ) .
When r = 0, we can rewrite the equation above as
c(f lift,N,B , −n) = −(N + 1) ( ( )
r B pm ′ − n N + rB (pm ′ − Nn) )
( )
)
pNm
+
(r ′ −n
B + r
N
B (N(pNm ′ − n))
(
)
= −(N + 1)c T N (j lift,B
m
), −n + c (j ′
Nm ′(pz)T N (ϑ B )(z), −n) .
Notice that j pNm ′(pz)T N (ϑ B )(z) = j lift,B[n]
Nm
(z) + j lift,B[n]−1
′
Nm
(z). Since this is true for all n ≥ 1
′
and the principal part uniquely determines a form in M !,+
f lift,N,B (z) = −(N + 1)T N
(
j lift,B
m ′
)
(z)
+ j lift,B[n]
Nm ′
1 (p) up to forms in M + 1 (p), we have
(z) + j lift,B[n]−1 (z) + g f (z)
for some g f (z) ∈ M + 1 (p) with integral linear combination of c(f, n)’s as Fourier coefficients.
Since T N is self-adjoint with respect to the regularized inner product and
T N (g ψ )(z) = (ψ([n]) + ψ([n]))g ψ (z)
for N prime with χ p (N) = 1, we can rewrite 〈f lift,N,B , g ψ 〉 reg as
〈f lift,N,B , g ψ 〉 reg = −(N + 1)(ψ([n]) + ψ([n]))〈j lift,B
m ′ , g ψ 〉 reg + ρ N,B,ψ (Nm ′ ) + 〈g f , g ψ 〉.
Nm ′