Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 1 / a

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- Solution a

Solution

The dimension of level-one cusp forms is zero when

k

is odd or

k < 12

. For even

k \geq 12

it is

dim S_{k} (Γ (1)) = {⌊ k /12 ⌋ - 1, ⌊ k /12 ⌋, k \equiv 2 (mod 12), k \neq \equiv 2 (mod 12) .

Equivalently, multiplication by the modular discriminant gives

S_{k} (Γ (1)) = Δ M_{k - 12} (Γ (1))

, and the displayed formula follows from the standard dimension formula for

M_{k - 12} (Γ (1))

.

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