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

MOCK-MODULAR FORMS OF WEIGHT ONE 19
For convenience, we write E(τ, s) for E 1 (τ, s). Kronecker’s first limit formula states that
(49) ζ(2s)E(τ, s) = π
s − 1 + 2π ( γ − log(2) − log( √ y|η(τ)| 2 ) ) + O(s − 1),
where γ is the Euler constant. Using (49) and Rankin-Selberg unfolding trick, we can relate
the inner product between dihedral newforms to logarithm of u A as follows.
Proposition 4.1. Let ψ, ψ ′ be characters of Cl(F ), with ψ non-trivial. When ψ ′ = ψ or ψ,
we have
(50) 〈g ψ , g ψ ′〉 = 1 ∑
ψ 2 (A) log |u A |.
12
Otherwise, 〈g ψ , g ψ ′〉 = 0.
A∈Cl(F )
Proof. Since p is prime, the Eisenstein series E p (z, s) has a simple pole at s = 1 with residue
, which is independent of z. So we have the relationship
3
π(p+1)
3
π(p + 1) · 〈g ψ, g ψ ′〉 = Res s=1 g ψ (z)g ψ ′(z)E p (z, s)y
∫Γ dxdy .
0 (p)\H
y 2
Now, we can use the Rankin-Selberg method to unfold the right hand side and obtain
∫
g ψ (z)g ψ ′(z)E p (z, s)y dxdy = Γ(s) ∑ r ψ (n)r ψ ′(n)
y 2 (4π) s n s
Γ 0 (p)\H
Let ρ ψ : Gal(Q/Q) → GL 2 (C) be the representation induced from ψ. Then it is also the
one attached to g ψ (z) via Deligne-Serre’s theorem. Up to Euler factors at p, the right hand
side is L(s, ρ ψ ⊗ ρ ψ ′), the L-function of the tensor product of the representations ρ ψ and ρ ψ ′.
From the character table of the dihedral group D 2H(p) , we see that
ρ ψ ⊗ ρ ψ ′ = ρ ψψ ′ ⊕ ρ ψψ ′.
So when ψ ′ ≠ ψ or ψ ′ , the L-function L(s, ρ ψ ⊗ρ ψ ′) is holomorphic at s = 1 and 〈g ψ , g ψ ′〉 = 0.
Otherwise, we have
∑ r ψ (n)r ψ (n)
= ζ(s)L(s, χ p)L(s, ρ ψ 2)
,
n s ζ(2s)(1 + p −s )
n≥1
Putting these together, we obtain
(51) 〈g ψ , g ψ 〉 = p
2π L(1, χ p)L(1, ρ 2 ψ 2).
From the theory of quadratic forms, we have
√ ∑
pL(s, ρψ 2) = ζ(2s) ψ 2 (A)E(τ A , s).
A∈Cl(F )
Since ψ is non-trivial and H(p) is odd, ψ 2 is non-trivial and (49) implies that
L(1, ρ ψ 2) = −√ 2π ∑
ψ 2 (A) log( √ y A |η(τ A )| 2 ),
p
A∈Cl(F )
Along with the class number formula for p > 3
L(1, χ p ) = 2πH(p)
w p
√ p
n≥1
= πH(p) √ p
,