Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 2 / a

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- Solution a

Solution

The Gamma 1 congruence subgroup is

Γ_{1} (N) = {(a c b d) \in S L_{2} (Z) : a \equiv d \equiv 1 (mod N), c \equiv 0 (mod N)} .

M_{k} (Γ_{1} (N))

consists of the holomorphic functions

f : h \to C

satisfying

f (γ τ) = (c τ + d)^{k} f (τ)

for every

γ \in Γ_{1} (N)

and which are holomorphic at a cusp for every cusp. Its subspace

S_{k} (Γ_{1} (N))

consists of the forms that vanish at every cusp.

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