Many conjectures about properties of automorphic forms on $\mathrm{GL}(2)$ can be formulated in the basic language of classical modular forms (i.e. Hecke forms that are holomorphic on $\mathbb{H}$ or nonholomorphic Maass forms). However, sometimes the proofs of these conjectures are best understood (or indeed only understood!) by first translating the problem to the adelic setting of automorphic representations of $\mathrm{GL}_2\left(\mathbb{A}_{\mathbb{Q}}\right)$.

An example is a recent paper of Templier disproving a folklore conjecture that $\|f\|_{\infty} \ll_{\varepsilon} N^{\varepsilon}$ for $f \in S_k(N,\chi)$, by (essentially) showing that the local newvector $W_p$ in the Whittaker model of the automorphic representation associated to a newform $f \in S_k^{*}(N,\chi)$ takes large values when $p^2 | N$.

My question is: what are other examples of adelic methods being used to solve problems in the classical theory of modular forms?