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

42 W. DUKE AND Y. LI
and we have
r + ψ (n) = −β ∑ A
ψ 2 (A) log |u(n, A)| = −β ′ ∑ A
ψ 2 (A) log |u ′ (n, A)|.
By the choice of β ′ and u ′ (n, A), we also have
(90)
∑
R p + (n) = r + A (n)
A∈Cl(F )
∑
= −β ′ log |u ′ (n, A)|
A∈Cl(F )
= − β′
2 log |N H/Q(u ′ (n, A))|.
This is an analogue of equation (9) for mock-modular forms as mentioned in the introduction.
□
8. Counting CM points
In preparation for the proof of Theorem 1.2, we will count the number of CM points on a
hyperbolic circle in terms of the representation numbers of positive definite binary quadratic
forms. As before, such a counting argument is needed to construct a Green’s function at
special points. This construction follows ideas of [23], but in the counting argument given
there one also sums over all classes of a given discriminant. Even when one discriminant is
prime, which we are assuming, their proof involves quite an intricate application of algebraic
number theory. Surprisingly, the refined version we need for a fixed class has a completely
elementary proof using the classical theory of composition of binary quadratic forms. It
has the further advantage that it applies without extra effort when the other discriminant
is not fundamental. Although we will not give details here, the argument adapts to give a
corresponding refinement of the level N case in [24].
Good references for the theory of binary quadratic forms are the books by Buell [3] and
Cox [14]. Let −d < 0 be a discriminant, not necessarily fundamental, and Q −d the set of
positive definite integral binary quadratic forms
Q(x, y) = ax 2 + bxy + cy 2
with −d = b 2 − 4ac, where a, b, c ∈ Z. For any Q ∈ Q −d the associated CM point is defined
to be
τ Q = −b + √ −d
∈ H,
2a
where H is the upper half plane. Clearly Q(τ Q , 1) = 0. The modular group Γ = PSL 2 (Z) acts
on Q ∈ Q −d by a linear change of variables, which induces linear fractional transformation
on τ Q . Let w Q be the number of stabilizers of Q. The number of equivalence classes of
quadratic forms is the Hurwitz class number H(d). Those classes represented by primitive
forms (those with gcd(a, b, c) = 1) comprise a finite abelian group under composition, the
class group. When −d is fundamental, this class group is canonically isomorphic to the ideal
class group of Q( √ −d) by sending [Q] to the class A ∈ Cl(Q( √ −d)) containing the fractional
ideal Z + Zτ Q . For primitive Q ∈ Q −d , let Q 2 denote a representative of the square class of
Q.
Associated to Q is the counting function
r Q (k) = 1 2 #{(x, y) ∈ Z2 | Q(x, y) = k},