Let $\mathcal L$ be the set of <lattices> in $\mathbb C$. The <lattice model of a modular form> is
$$
V_k=\{F:\mathcal L\to\mathbb C:F(\lambda\Lambda)=\lambda^{-k}F(\Lambda)
\text{ for every }\lambda\in\mathbb C^\times\}.
$$
For the normalization used here, the $n$th <Hecke operator on lattice functions> is
$$
(T_nF)(\Lambda)
=n^{k-1}\sum_{\Lambda'\supset\Lambda\atop[\Lambda':\Lambda]=n}F(\Lambda').
$$
There are finitely many such superlattices, and scaling gives a bijection between those above $\Lambda$ and those above $\lambda\Lambda$, so $T_nF$ again has weight $k$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 4 / a / i / Solution

{title2=$V_k$}

Let $\mathcal L$ be the set of complex lattices. A weight-$k$ lattice function satisfies $F(\lambda\Lambda)=\lambda^{-k}F(\Lambda)$. Evaluating at $\Lambda_\tau=\mathbb Z\tau+\mathbb Z$ identifies such functions with weight-$k$ modular-invariant functions on the upper half-plane.


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