The orbit method in number theory through the sup-norm problem for $\mathrm{GL}(2)$

The orbit method in its quantitative form due to Nelson and Venkatesh has played a central role in recent advances in the analytic theory of higher rank $L$-functions. The goal of this note is to explain how the method can be applied to the sup-norm problem for automorphic forms on $\mathrm{PGL}(2)$. Doing so, we prove a new hybrid bound for newforms $\phi$ of large prime-power level $N=p^{4n}$ and large eigenvalue $\lambda$. It states that $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}{\ll }_p(\lambda N{)}^{5\mathrm{/}24+\epsilon }$, recovering the result of Iwaniec and Sarnak spectrally and improving the local bound in the depth aspect for the first time in this non-compact setting. We also provide an exposition of the microlocal tools used, illustrating and motivating the theory through the classical case of $\mathrm{PGL}(2)$, following notes and lectures of Nelson and Venkatesh.