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Mathematical Tripos Part III: Modular Forms Exam Paper, Exams of Mathematics

A past exam paper from the university of cambridge's mathematical tripos part iii course, focusing on modular forms. The paper includes questions on topics such as lattices, modular invariants, hecke operators, and cusp forms. Students are required to use results and formulas from the theory of modular forms to solve problems. The paper also includes instructions for the exam, such as the duration, stationery requirements, and special instructions.

Typology: Exams

2012/2013

Uploaded on 02/28/2013

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MATHEMATICAL TRIPOS Part III
Monday 11 June 2007 1.30 to 4.30
PAPER 29
MODULAR FORMS
Attempt FOUR questions.
There are SIX questions in total.
The questions carry equal weight.
For any σ=a b
c d SL2(Z)we write f|[σ]k(τ) = ( +d)kf(σ(τ)).
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Cover sheet None
Treasury Tag
Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
pf3
pf4

Partial preview of the text

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MATHEMATICAL TRIPOS Part III

Monday 11 June 2007 1.30 to 4.

PAPER 29

MODULAR FORMS

Attempt FOUR questions.

There are SIX questions in total.

The questions carry equal weight.

For any σ =

a b c d

∈ SL 2 (Z) we write f |[σ]k(τ ) = (cτ + d)−kf (σ(τ )).

STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS

Cover sheet None Treasury Tag Script paper

You may not start to read the questions

printed on the subsequent pages until

instructed to do so by the Invigilator.

(a) Show that mapping τ ∈ H to the lattice Λτ := Zτ + Z induces a bijection between SL 2 (Z)\H and the set of lattices in C up to homothety.

(b) Prove that the modular invariant j : H → C induces a bijection SL 2 (Z)\H → C. [You may assume the formula ord∞(f )+ 12 ordi(f )+ 13 ordρ(f )+

τ 6 =i,ρ ordτ^ (f^ ) =^

k 6 for non-zero f ∈ M 2 k(SL 2 (Z)).]

(c) Let Λ, Λ′^ ⊂ C be lattices satisfying G 4 (Λ) = G 4 (Λ′) and G 6 (Λ) = G 6 (Λ′). Prove that Λ = Λ′.

2 (a) Define the topology and complex structure of X(1) = SL 2 (Z)\H∗^ and prove that X(1) is compact.

(b) Prove using facts about compact Riemann surfaces that the space M 2 k(SL 2 (Z)) is finite dimensional.

3 Let E 2 (τ ) = 1 − 24

n=1 σ^1 (n)q

n (^) for q = e 2 πiτ (^).

(a) Using the relation E 2 (− 1 /τ ) = τ 2 E 2 (τ ) + (^6) πiτ prove that

F (τ ) = q

∏^ ∞

n=

(1 − qn)^24

lies in S 12 (SL 2 (Z)).

(b) Let Θ = q (^) dqd = (^21) πidτd. Show that, for every f ∈ Mk(SL 2 (Z)),

g = (Θ −

k 12

E 2 )f ∈ Mk+2(SL 2 (Z))

and that f ∈ Sk(SL 2 (Z)) if and only if g ∈ Sk+2(SL 2 (Z)).

(c) Show that the coefficients τ (n) of F (τ ) = q

n=1(1^ −^ q

n) (^24) = ∑∞ n=1 τ^ (n)q

n (^) satisfy

(1 − n)τ (n) = 24

n∑− 1

`=

σ 1 ()τ (n −)

and τ (n) ≡ nσ 5 (n) (mod 5).

[You may use without proof that dimM 8 (SL 2 (Z)) = dimS 12 (SL 2 (Z)) = 1 and dimS 14 (SL 2 (Z))) = 0.]

Paper 29

6 (a) Define the space Sk(Γ) of cusp forms of weight k with respect to a congruence subgroup Γ ⊂ SL 2 (Z).

(b) For f : H → C let φ(τ ) = |f (τ )|(Im(τ ))

k 2

. Show that for σ ∈ SL 2 (Z)

φ(σ(τ )) = |f |[σ]k(τ )|(Im(τ ))

k 2 .

(c) Prove that a function f : H → C is an element of Sk(Γ) if and only if the following three conditions hold:

i. f is meromorphic on H,

ii. f |[γ]k = f for all γ ∈ Γ, iii. f (τ )(Im(τ ))k/^2 is bounded on H.

END OF PAPER

Paper 29