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

For $\tau \in \mathfrak{h}$, put ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ and defineIf $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$, thenso $f_F(\gamma \tau )=(c\tau +d{)}^k{f}_F(\tau )$. Conversely, if $f$ has this invariance and $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ with $\mathrm{Im}({\omega }_1/{\omega }_2)>0$, defineThe modular transformation law makes this independent of the oriented basis. These constructions are inverse, identifying $V_k$ with $W_k$.

Transporting $T_n$ through this identification givesThis defines the Hecke operator on $W_k$.