 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-29/6/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Write

τ = x + i y

and

σ = (a c b d)

. Since

a d - b c = 1

Im (σ τ) = \frac{y}{∣ c τ + d ∣ ^{2}} .

ϕ (σ τ) = ∣ f (σ τ) ∣ (\frac{y}{∣ c τ + d ∣ ^{2}})^{k /2} = ∣ (c τ + d)^{- k} f (σ τ) ∣ y^{k /2} = ∣ f ∣_{k} σ (τ) ∣ y^{k /2} .

This computation applies to an arbitrary function; modularity is needed only to deduce

Γ

-invariance of

ϕ

. The exponent is $k /2$ , as printed in the PDF, rather than the converted TeX's

1/2

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