Modular Forms: Problem Sheet 3

ADVERTISEMENT

Modular Forms: Problem Sheet 3
23 February 2016
1. (a) Prove the formula
n 1
21
10
5040
σ
(n) =
σ
(n)
σ
(n) +
σ
(j)σ
(n
j) for all n
.
Z
9
5
3
3
5
>0
11
11
11
j=1
(b) Find similar expressions for σ
in terms of σ
and σ
, and in terms of
13
3
9
σ
and σ
.
5
7
2. (a) Find rational numbers λ and µ such that
3
∆ = λE
+ µE
.
12
4
(b) Let τ (n) be the n-th coefficient in the q-expansion of ∆, so that
n
∆ =
τ (n)q
.
n=1
Prove Ramanujan’s congruence:
τ (n)
σ
(n) (mod 691).
11
3. Show that the ring C[E
, E
, E
] is closed under differentiation.
2
4
6
4. (a) Show that the modular functions (for SL
(Z)) form a field F (with ad-
2
dition and multiplication defined pointwise).
(b) Prove that F = C(j) and that j is transcendental over C.
5. Consider the modular function j : H
C.
(a) Show that j(i) = 1728 and j(ρ) = 0 (where ρ = exp(2πi/3)).
(b) Let z
(the standard fundamental domain for SL
(Z)). Prove:
2
(z lies on the boundary of
or
z = 0)
j(z)
R.
(c) Show that j : SL
(Z) H
C given by j([z]) := j(z) is well-defined and
2
prove that j is bijective.
(Here [z] denotes the orbit of z under the action of SL
(Z).)
2
(d) Prove the converse to part (b).
a
b
6. (a) Show that M
is spanned by all E
E
with a, b
and 4a + 6b = k.
Z
k
0
4
6
(b) Show that E
and E
are algebraically independent over C.
4
6
We remark that this exercise shows that the ring of modular forms (for
SL
(Z)) M :=
M
is isomorphic to the ring of polynomials over C in two
2
k
k Z
variables C[x, y] with isomorphism C[x, y]
M given by (x, y)
(E
, E
).
4
6
(If we grade the rings by assigning grade k to a modular form of weight k
and grades 4 and 6 to x and y respectively, we get an isomorphism of graded
rings.)
1

ADVERTISEMENT

00 votes

Related Articles

Related forms

Related Categories

Parent category: Education
Go