MA342J: Introduction to Modular Forms
Study week assignment
1. (“Eisenstein series of weight 2”) The series
X X E2 (z) = 1 − 24
d qn
n≥1
d|n
is convergent for |q| < 1 and therefore defines a holomorphic function in the
upper half-plane. Find in literature (one can try [1-2-3]) and understand
the proof of the following statement.
Theorem. The nonholomorphic function
E2∗ (z) = E2 (z) −
3
πIm(z)
is equivariant under SL(2, Z) in weight 2, i.e.
1
∗ az + b
E
= E2∗ (z) .
2
(cz + d)2
cz + d
2. (“Jacobi’s Theorem”) Prove the identity
Y
∆ = q
(1 − q n )24 ,
n≥1
1
where ∆ = 1728
(E43 − E62 ) is the cusp form of weight 12 constructed in
class, by showing that the product in the r.h.s. is a modular form. Let us
denote this product by
Y
˜ = q
∆
(1 − q n )24 .
n≥1
˜ is a q-series it obviously satisfies ∆(z
˜ + 1) = ∆(z).
˜
Since ∆
It would be
sufficient to prove the functional equation
1 ˜ 1
˜
∆ −
= ∆(z)
.
z 12
z
To this end do the following steps:
a) Prove that
X
1 d
˜ = q d log ∆
˜ = q d log q + 24
log ∆
log(1 − q n ))
2πi dz
dq
dq
n≥1
is equal to E2 .
˜ satisfies
b) Combine a) with the theorem from question 1 to show that ∆
the above functional equation.
1