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The theory of modular forms and hecke operators for the modular group sl2(z). It includes definitions, formulas, and proofs of various properties. Topics covered include the spaces mk and sk, multiplication by ∆, dirichlet series l(g, s), and the poisson summation formula.
Typology: Exams
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Wednesday 4 June 2003 9 to 12
Attempt FOUR questions. There are five questions in total. The questions carry equal weight.
1 Define the spaces Mk, Sk of modular and cusp forms of weight k for the modular group SL 2 (Z). Show that multiplication by ∆ = (E^34 − E^26 )/1728 is an isomorphism between Mk and Sk+12, and obtain the dimension formula for even k ≥ 0:
dim Mk =
k 12
if k ≡ 2 (mod 12) [ k 12
otherwise.
Show also that every modular form on SL 2 (Z) can be expressed as a polynomial in E 4 and E 6.
[You may assume the formula ∑
τ 6 =i,ρ
vτ (f ) +
2 vi(f^ ) +
3 vρ(f^ ) +^ v∞(f^ ) =^
k 12
for non-zero f ∈ Mk.]
ii) Show that if E 4 ∆ =
n≥ 1 cnq
n (^) then cn ≡ σ 15 (n) (mod 3617).
[The q-expansion of E 16 is 1 +
n=
σ 15 (n)qn. ]
2 Write an account of the theory of Hecke operators for modular forms on SL 2 (Z).
Paper 24
4 (i) State and prove the Poisson summation formula, and use it to show that the theta function
ϑ 00 (z, τ ) =
n=−∞
qn
(^2) / 2 tn^ (q = e^2 πiτ^ , t = e^2 πiz^ )
satisfies
ϑ 00 (z/τ, − 1 /τ ) =
( (^) τ i
eπiz
(^2) /τ ϑ 00 (z, τ )
(ii) Show that
ϑ 00 (z, τ )ϑ 01 (z, τ )ϑ 10 (z, τ )ϑ 11 (z, τ ) ϑ 11 (2z, τ ) =
2 ϑ^00 (0, τ^ )ϑ^01 (0, τ^ )ϑ^10 (0, τ^ )
where ϑαβ (z, τ ) = iαβ^ qα/^8 tα/^2 ϑ 00 (z + ατ^ +^ β 2
, τ ).
[The transformation formula for ϑαβ is
(−1)αϑαβ (z + 1, τ ) = ϑαβ (z, τ ) = (−1)β^ q^1 /^2 t ϑαβ (z + τ, τ ). ]
5 i) What does it mean to say that a function on the upper half plane is modular of weight k on a subgroup Γ ⊂ SL 2 (Z)? Show that if f (τ ) is modular of weight k on SL 2 (Z) then for any positive integer N , f (N τ ) is modular of weight k on
a b c d
∣∣ c ≡ 0 (mod N )
ii) Let E 2 (τ ) = 1 − 24
n≥ 1 σ^1 (n)qn. Show that
E 2 (− 1 /τ ) = τ 2 E 2 (τ ) +^6 τ πi
Deduce that E 2 ∗ (τ ) = E 2 (τ ) − 3 πy
is modular of weight 2 on SL 2 (Z), where y = Im(τ ).
[You may assume that ∆ has the product expansion ∆(τ ) = q
(1 − qn)^24 , and that P SL 2 (Z) is generated by the transformations τ 7 → τ + 1, τ 7 → − 1 /τ .]
iii) Let N > 1 and c(M ) ∈ C be given for each M |N. Show that ∑
M |N
c(M )E 2 (M τ )
is modular of weight 2 on Γ 0 (N ) if and only if
M −^1 c(M ) = 0.
Paper 24
