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

MOCK-MODULAR FORMS OF WEIGHT ONE 17
Since ξ 1 commutes with the action of U p and W p and conjugates their coefficients, it commutes
with the operator pr ɛ as follows
(41) ξ 1 (pr ɛ (f)) = pr −ɛ (ξ 1 (f)).
The operator pr −ɛ is the identity when restricted to S1 −ɛ (p). So the preimage of S1 −ɛ (p) under
ξ 1 lies in H1(p) ɛ and the map ξ 1 : H1(p) ɛ → S1 −ɛ (p) is surjective for ɛ = ±1.
Now if f ∈ S 1 (p) is a dihedral newform, then it is a linear combination of theta series, whose
n th coefficient is zero if χ p (n) = −1. So f ∈ S 1 + (p) by definition. If f(z) = ∑ n≥1
c(f, n)qn
is a octahedral or icosahedral newform, then f(z) and f(z) := f(z) = ∑ n≥1 c(f, n)qn are
linearly independent newforms. When l ≠ p is a prime number, we have the relationship
(42) c(f, l) = χ p (l)c(f, l).
This is a consequence of the formula of the adjoint of the l th Hecke operator with respect to
the Petersson inner product [34, p.21]. By the definition of S1(p), ɛ we have f + f ∈ S 1 + (p)
and f − f ∈ S1 − (p). Since the number of octahedral and icosahedral newforms is set to be
2d − , we obtain the formulae for the dimensions.
□
Remark: The spaces S ɛ 1(p) should be compared to the spaces M + and M − in (9.1.2) in
[36]. If all octahedral and icosahedral newforms f(z) ∈ S 1 (p) satisfy
(43) (f| 1 W p )(z) = −if(z),
then M + is the span of the weight one Eisenstein series and S + 1 (p) and M − = S − 1 (p).
Besides compatibility with ξ 1 , the decomposition also behaves nicely with respect to the
regularized inner product. As a consequence of Lemma 3.1, we have the following proposition.
Proposition 3.4. Let f(z) ∈ M !,ɛ
1 (p), g(z) ∈ S ɛ′
1 (p) and ˜g(z) ∈ M −ɛ′
1 (p) with shadow g(z)
and Fourier expansion ∑ n∈Z c+ (n)q n . If ɛ = ɛ ′ , then
(44) 〈f, g〉 reg = ∑ n∈Z
δ(n)c(f, −n)c + (n).
Here δ(n) is 2 if p|n and 1 otherwise. If ɛ ≠ ɛ ′ , then 〈f, g〉 reg = 0.
Proof. Let ĝ(z) ∈ H −ɛ′
1 (p) be the harmonic Maass form associated to ˜g(z). Then both
assertions are immediate consequences of Lemma 3.1 and
(f| 1 W p )(z) = −iɛ(f| 1 U p )(z), (ĝ| 1 W p )(z) = −iɛ ′ (ĝ| 1 U p )(z).
Since the adjoint of W p (resp. U p ) is W −1
p
= −W p (resp. W p U p Wp
−1 ) with respect to the
regularized inner product, it is easy to check that pr ɛ is self-adjoint, which also proves the
second claim.
□
When g(z) above is zero, i.e. ˜g(z) is modular, equation (44) reduces to an orthogonal
relationship between Fourier coefficients of weakly holomorphic modular forms. In particular,
we can take ˜g(z) to be cusp forms and obtain relations satisfied by {c(f, −n) : n ≥ 1}. These
relations turn out to characterize the space M !,ɛ
1 (p), which gives a nice characterization of
the space M ɛ 1(p).
Proposition 3.5. Let g(z) ∈ S1(p). ɛ Then there exists a mock-modular form ˜g(z) ∈ M −ɛ
1 (p)
with shadow g(z) and prescribed principal part ∑ n