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

40 W. DUKE AND Y. LI
(i) d(n) = 0 for all n with χ p (n) = 1,
(ii) ∑ n∈Z c(H, n)d(n)δ(n) = 0 for all H(z) = ∑ n∈Z c(H, n)qn ∈ M !,+,lift
1 (p),
(iii) D(z) ∉ S1 − (p).
In particular for H(z) = f lift,N,B (z), condition (ii) can be rewritten as
(89) ∑ m∈Z
c(f, −m) ∑ k∈Z
Also, when H(z) = j lift,B[n]
m ′
∑
δ(k)d(k)
δ(k)d(k)r B (pm − Nk) = N ∑ m ′ ∈Z
k∈Z
So equation (89) can be written as
∑
c(f, −Nm ′ ) ∑ k∈Z
(z) + j lift,B[n]−1
m
(z), we have
(r ′
( )
)
B + r B (pNm ′ − Nk) = 0.
pm ′ −k
N
m∈Z
δ N (m)c(f, −m) ∑ k∈Z
δ(k)d(k)r B (pm − Nk) = 0.
δ(k)d(k)r B
(
By Lemma 7.1, D(z) ∈ S − 1 (p). This contradicts condition (iii) and we must have
M !,+,lift
1 (p) = M !,+
1 (p).
)
pm ′ −k
.
N
With Proposition 6.1 and Lemma 7.1, we are ready to give the proof of Theorem 1.1.
Proof of Theorem 1.1. Denote the dimension and the q-echelon basis of S − 1 (p) by d − and the
basis by
{h t (z) ∈ S − 1 (p) : h t (z) = q nt + O(q n d − +1
), 1 ≤ t ≤ d − }.
Choose a mock-modular form ˜g ψ (z) = ∑ n≥n 0
r + ψ (n)qn such that it has a fixed principal part as
in Proposition 4.2 and r + ψ (n t) satisfies conditions (ii) and (iii) in Theorem 1.1 for 1 ≤ t ≤ d − .
Note that if S1 − (p) = ∅, then there is only one ˜g ψ (z) with a fixed principal part. We claim
that this ˜g ψ (z) is a desired choice.
Consider all f(z) ∈ M !,new
0 (N j ), 1 ≤ j ≤ (p − 1)/2 with rational Fourier coefficients and
j m (z) for all m ≥ 0. They generate the complex vector space M !,new
0 (N j ) and M 0(1) ! respectively.
They also give rise to equation (86) and equation (66), which form a system of equations
satisfied by {r + ψ (n) : n ≥ n 0}. Suppose there is another set of solutions {c + (n) : n ≥ n 0 }
with r + ψ (n) = c+ (n) whenever n ≤ 0, χ p (n) = 1, or n = n t for t = 1, . . . , d − . Then by Lemma
7.1, the power series
D(z) := ∑ n≥1
(r + ψ (n) − c+ (n))q n
is a cusp form in S1 − (p). However, since r + ψ (n t) = c + (n t ) for all 1 ≤ t ≤ d − , the cusp
form D(z) must be zero in S1 − (p). So with {r + ψ (n) : n ≤ 0 or n = n t, 1 ≤ t ≤ d − } fixed,
{r + ψ (n) : n ≥ d − + 1} is the unique solution to this system of equations. From Corollary
7.2 and Proposition 3.5, it is clear that r + ψ
(n) can be solved from this system of equations,
i.e. the r + ψ
(n)’s are a rational linear combination of these regularized inner products. By
Corollary 5.2 and Proposition 6.1, we can write
r + ψ (n) = −β ∑
n ψ 2 (A) log |u(n, A)|,
A∈Cl(F )
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