14 1. BASIC FACTS
Theorem 1.49. In the notation above, the following are true:
(1) If k ∈
dimC (Mk(Γ0(4), ψk)) =
if k 0,
+ [k/2] if k ≥ 0,
dimC (Sk(Γ0(4), ψk)) =
if k ≤ 2,
[k/2] − 1 if k 2 and k ∈ 2Z,
− 2 if k 2 and k ∈ 2Z.
(2) As a graded algebra, we have
Mk(Γ0(4), ψk) = C[F, θ].
We also recall Cohen’s Eisenstein series [C1]; these are explicit half-integral
weight modular forms which generalize the classical Eisenstein series Ek(z). Their
Fourier coeﬃcients are given by generalized Bernoulli numbers for quadratic char-
acters. These half-integral weight modular forms will be important in Section 8.4.
Here we recall their definition; first we recall the generalized Bernoulli numbers.
Definition 1.50. Let χ be a nontrivial Dirichlet character modulo m. The
generalized Bernoulli numbers B(n, χ) are defined by the generating function
B(n, χ) ·
emt − 1
If χ is a Dirichlet character, then its Dirichlet L-function is given by
(1.19) L(s, χ) =
Generalized Bernoulli numbers give the values of Dirichlet L-functions at nonposi-
tive integers (for example, see Proposition 16.6.2 of [IR]).
Proposition 1.51. If k is a positive integer and χ is a nontrivial Dirichlet
L(1 − k, χ) = −
If D is a fundamental discriminant (i.e. the discriminant of a quadratic number
field), then let
be the Kronecker character for Q(
D). Using these Dirichlet characters, we now
define Cohen’s Eisenstein series.
Fix an integer r ≥ 2. If (−1)rN ≡ 0, 1 (mod 4), then let H(r, N) := 0. If
N = 0, then let
H(r, 0) := ζ(1 − 2r) = −