14 1. BASIC FACTS

Theorem 1.49. In the notation above, the following are true:

(1) If k ∈

1

2

Z, then

dimC (Mk(Γ0(4), ψk)) =

0

1

if k 0,

+ [k/2] if k ≥ 0,

dimC (Sk(Γ0(4), ψk)) =

0

[k/2]

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

k∈

1

2

Z

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

∞

n=0

B(n, χ) ·

tn

n!

=

m−1

a=1

χ(a)teat

emt − 1

.

If χ is a Dirichlet character, then its Dirichlet L-function is given by

(1.19) L(s, χ) =

∞

n=1

χ(n)

ns

.

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

character, then

L(1 − k, χ) = −

B(k, χ)

k

.

If D is a fundamental discriminant (i.e. the discriminant of a quadratic number

field), then let

χD =

D

•

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) = −

B2r

2r

.