# Adelic methods for classical modular forms

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?

• Not quite adélic but representation-theoretic are some of the papers of Bernstein and Reznikov (e.g., "Analytic continuation of representations and estimates of automorphic forms", Annals of Math. 150 (1999), 329-352). In fact, representation-theoretic insights and adélic insights are often conflated because both were introduced around the same time, but sometimes the first is enough. – Denis Chaperon de Lauzières Dec 2 '13 at 18:18

A standard example of a theory that can be understood without the adelic setting but are better understood with it is the Atkin-Lehner theory of new forms. Atkin and Lehner did prove it without any reference to the adelic settings, using instead heavily the $q$-expansions of modular forms. But the theory is cumbersome to state and to prove in the standard setting, while it becomes much simpler and more natural in the adelic point of view, as was emphasized by Jacquet-Langlands.
• Yes, and one should mention W. Casselman's contribution to the modernization of Atkin-Lehner for $GL(2)$... – paul garrett Dec 3 '13 at 3:00
This answer is partly inspired by Joël's, but I think it's worth mentioning that the method of newforms is much coarser then theory of types. Modular Hecke cusp forms for fixed weight but different level often share the "same" L-function. The L-function does depend significantly on the irreducible cupsidal automorphic representation, not so much upon the precise vector given. If you are willing to use the theory of types and the corresponding congruence representation-valued forms, you are able to see where redundant things are happening. E.g. given a supercuspidal representation of $SL_2(F_p)$, you can inflate it to $\tau_p$ of $SL_2(Z_p)$ and $\tau$ of $SL_2(Z)$. If you study the corresponding $\tau$ vector-valued forms, the corresponding local factor of the supercuspidal representations are precisely the supercuspidal given by the induced of $\tau_p$ up to $SL_2(Q_p)$. These also turn up as newvectors somewhere, but they can't be separated from the other stuff happening there anymore. Note that the corresponding $L$-factor is constant.