10 1. BASIC FACTS

1.2.3. Dimension formulas. Proposition 1.25 provides formulas for the di-

mensions of Mk and Sk. Using the Riemann-Roch Theorem, Cohen and Oesterl´e

[COe] explicitly computed further dimension formulas which we record here be-

cause of their utility. To state these formulas, suppose that k is an integer, and

that χ is a Dirichlet character modulo N for which χ(−1) =

(−1)k.

If p | N is

prime, then let rp (resp. sp) denote the power of p dividing N (resp. the conductor

of χ). Define the integer λ(rp, sp, p) by

(1.11) λ(rp, sp, p) :=

pr

+

pr −1

if 2sp ≤ rp = 2r ,

2pr if 2sp ≤ rp = 2r + 1,

2prp−sp

if 2sp rp.

In addition, define rational numbers νk and µk by

νk :=

0

if k is odd,

−1/4 if k ≡ 2 (mod 4),

1/4 if k ≡ 0 (mod 4),

µk :=

0

if k ≡ 1 (mod 3),

−1/3 if k ≡ 2 (mod 3),

1/3 if k ≡ 0 (mod 3).

(1.12)

In this notation, we have the following dimension formulas.

Theorem 1.34. If k is an integer and χ is a Dirichlet character modulo N for

which χ(−1) =

(−1)k,

then

dimC(Sk(Γ0(N), χ)) − dimC(M2−k(Γ0(N), χ))

=

(k − 1)N

12

·

p|N

(1 +

p−1)

−

1

2

p|N

λ(rp, sp, p) + νk

x (mod N),

x2+1≡0

(mod N)

χ(x) + µk

x (mod N),

x2+x+1≡0

(mod N)

χ(x),

where p denotes a prime divisor of N (note. empty products are taken to be 1).

Remark 1.35. If k 2, then dimC(M2−k(Γ0(N), χ)) = 0. Hence the left

hand of side of Theorem 1.34 reduces to dimC(Sk(Γ0(N), χ)). A similar argument

applies when k = 2, and the result depends on whether χ is trivial. If k ≤ 0, then

dimC(Sk(Γ0(N), χ)) = 0. In these cases, the left hand side of Theorem 1.34 reduces

to − dimC(M2−k(Γ0(N), χ)).

1.3. Half-integral weight modular forms

Although the study of half-integral weight modular forms has its origins in

the classic works of Euler, Gauss and Jacobi (among others), many of their most