For $\tau\in\mathfrak h$, put $\Lambda_\tau=\mathbb Z\tau+\mathbb Z$ and define
$$
f_F(\tau)=F(\Lambda_\tau).
$$
If $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma(1)$, then
$$
\Lambda_{\gamma\tau}=(c\tau+d)^{-1}\Lambda_\tau,
$$
so $f_F(\gamma\tau)=(c\tau+d)^kf_F(\tau)$. Conversely, if $f$ has this invariance and $\Lambda=\mathbb Z\omega_1+\mathbb Z\omega_2$ with $\operatorname{Im}(\omega_1/\omega_2)>0$, define
$$
F_f(\Lambda)=\omega_2^{-k}f(\omega_1/\omega_2).
$$
The 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 gives
$$
(T_nf)(\tau)
=n^{k-1}\sum_{ad=n\atop d>0}d^{-k}
\sum_{b\bmod d}f\left(\frac{a\tau+b}{d}\right).
$$
This defines the Hecke operator on $W_k$.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-137/4/a/ii/solution