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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
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Monday 11 June 2007 1.30 to 4.
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 )).
Cover sheet None Treasury Tag Script paper
(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.
Paper 29
