For a cusp form $f$, the <invariant norm of a modular form>
$$
y^{k/2}|f(x+iy)|
$$
is modular invariant, bounded on $\mathcal F$, and tends to zero at its cusp. Part a therefore makes it bounded throughout $\mathfrak h$: $|f(x+iy)|\leq Cy^{-k/2}$.

The correct PDF expansion is $f(\tau)=\sum_{n\geq1}a_n(f)q^n$. Fourier inversion gives
$$
a_n(f)e^{-2\pi ny}=\int_0^1f(x+iy)e^{-2\pi inx}\,dx,
$$
and hence
$$
|a_n(f)|\leq Ce^{2\pi ny}y^{-k/2}.
$$
Choosing $y=1/n$ proves the <Fourier coefficient bound for a cusp form> $|a_n(f)|\leq C'n^{k/2}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-137/2/b/solution