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

MOCK-MODULAR FORMS OF WEIGHT ONE 13
i.e. for any cusp form h(z) ∈ S 1 (M, ν), there exists ˜h(z) ∈ M 1 (M, ν) with shadow h(z).
Proof. When k = 1, equation (2) becomes ∆ 1 = ξ 1 ◦ ξ 1 . So the Poincaré series P m (z, s)
satisfies
(26) ∆ 1 (P m (z, s)) = ( s − 1 2) 2
Pm (z, s)
when Re(s) > 1. Since the difference between both sides is holomorphic in s and P m (z, s)
can be analytically continued to Re(s) > 0 as in Proposition 2.1, equation (26) is valid for
Re(s) > 0. At s = 1/2, suppose P m (z, s) has the following Taylor series expansion in s
P m (z, s) = g −1 (z) ( s − 1 2) −1
+ g0 (z) + g 1 (z) ( s − 1 2)
+ Oz
( (s
−
1
2
) 2
)
,
with g j (z) ∈ F 1 (M, ν) real-analytic for j = −1, 0, 1. Since ξ 1 commutes with the slash
operator and ∆ 1 (g 1 (z)) = ξ1(g 2 1 (z)) = g −1 (z), we have ξ 1 (g 1 (z)) ∈ F 1 (M, ν) is real-analytic
and a preimage of g −1 (z) under ξ 1 .
By considering the Fourier expansion of g 1 (z), we know that if g 1 (z) has at most linear
exponential growth near the cusps, then ξ 1 (g 1 (z)) has the same property. Suppose Q m (z, s)
has the Laurent expansion
Q m (z, s) = g −1 (z) ( s − 1 2) −1
+ g
′
0 (z) + g ′ 1(z) ( s − 1 2)
+ Oz
( (s
−
1
2
near s = 1/2. Then from the spectral expansion of Q m (z, s), i.e. y −1/2 (D(z, s) + C(z, s))
as in (20), it is not hard to see that g 1(z) ′ has at most linear exponential growth near the
cusps. The difference P m (z, s) − Q m (z, s) is a Poincaré series defined for Re(s) > 0. So the
coefficient of (s − 1/2) of its Laurent series expansion around s = 1/2, say g 1(z), ′′ has at most
linear exponential growth near the cusps. That means the sum of g 1(z) ′ and g 1(z), ′′ which is
g 1 (z) by analytic continuation, also has this property. Thus, ξ 1 (g 1 (z)) is in H 1 (M, ν) and a
preimage of g −1 (z) ∈ S 1 (M, ν) under ξ 1 . Since the functions {Res s=1/2 P m (z, s) : m ≥ 1} span
the space S 1 (M, ν), the map ξ 1 : H 1 (M, ν) → S 1 (M, ν) is surjective.
□
3. Regularized Inner Products and the Weight One Plus Space
From now on, we fix M = p to be a prime number congruent to 3 modulo 4 and ν = χ p . The
spaces H 1(p, ! χ p ), H 1 (p, χ p ), M 1(p, ! χ p ), M 1 (p, χ p ), S 1 (p, χ p ), M ! 1(p, χ p ), M 1 (p, χ p ) are the same
as before, and we will drop χ p in these notations. In this section, we will relate the regularized
inner products between g(z) ∈ S 1 (p) and f(z) ∈ M 1(p) ! to linear combinations of coefficients
of a mock-modular form ˜g(z), whose shadow is g(z), via Stokes’ theorem. This technique
has been used in [9], [11] and [18]. Then, we will split up the space M ! 1(p) canonically into
the plus and minus spaces to facilitate further analysis. Whenever convenient, we will switch
between M ! 1(p) and H 1(p).
!
Given f(z) ∈ M 1(p) ! and g(z) ∈ S 1 (p), the usual Petersson inner product 〈f, g〉 can be
regularized as follows. Since p is prime, ( Γ 0 (p) has two inequivalent cusps, 0 and ∞. They
−1/
are related by the scaling matrix σ 0 =
√ p
√ p
). Take a fundamental domain of Γ 0 (p)\H,
cut off the portion with Im(z) > Y for a large Y and intersect it with its translate under σ 0 .
We will call this the truncated fundamental domain F(Y ). Now, define the regularized inner
product by
(27) 〈f, g〉 reg := lim
Y →∞
∫
F(Y )
f(z)g(z)y dxdy
y 2 .
) 2
)