1.2. INTEGER WEIGHT MODULAR FORMS 7
Theorem 1.23. If k 4 is even, then Mk is generated by monomials of the
form
E4(z)aE6(z)b,
where a, b 0 and 4a + 6b = k.
Using Theorem 1.23 to compute dimensions, Proposition 1.19 implies that
Mk = CEk (z)
for k {4, 6, 8, 10, 14}. As a consequence, we obtain elementary identities such as
E8(z) =
E4(z)2,
E10(z) = E4(z)E6(z),
E14(z) =
E4(z)2E6(z).
The weight k = 12 is the smallest integer for which Sk = {0}.
Definition 1.24. The Delta-function is the unique cusp form of weight 12 on
SL2(Z) normalized so that its leading Fourier coefficient equals 1. In terms of E4(z)
and E6(z), we have
∆(z) :=
E4(z)3

E6(z)2
1728
= q
24q2
+
252q3
··· Z[[q]].
The map
Ψk : Mk Sk+12
defined by Ψk(f(z)) := f(z)∆(z) is an isomorphism. Therefore, Theorem 1.23
immediately gives the following dimension formulas for Mk and Sk.
Proposition 1.25. If k 4 is even, then
dimC(Sk) = dimC(Mk) 1,
and
dimC(Mk) =

[k/12]



+ 1 if k 2 (mod 12),
[k/12] if k 2 (mod 12).
Modular functions on SL2(Z) are also simple to describe in terms of Eisenstein
series. To observe this we recall the modular j-function.
Definition 1.26. The modular j-function j(z) is defined by
j(z) :=
E4(z)3
∆(z)
=
q−1
+ 744 + 196884q +
21493760q2
+ ··· .
Remark 1.27. In terms of E6(z) and ∆(z), it also turns out that
j(z) 1728 =
E6(z)2
∆(z)
.
Previous Page Next Page