Let \(B/\mathbb{Q}\) be a definite quaternion algebra, and \(\mathcal{H}\) universal the Hecke algebra acting on it. To each idempotent \(e\) of the \(\overline{\mathbb{Q}}\)-vector space of compactly supported \(\mathcal{C}^\infty\)-functions on the adelic points of the algebraic group of invertible elements of \(B\), one associates an eigenvariety \(\mathcal{D}(e)\). The \(\overline{\mathbb{Q}}_p\)-points of \(\mathcal{D}(e)\) embeds into the homomorphisms from \(\mathcal{H}\) to \(\overline{\mathbb{Q}}_p\) times the \(\overline{\mathbb{Q}}_p\)-valued points of the weight space \(\mathcal{W}=\mathrm{Hom}_\mathrm{cont}(\mathbb{Z}_p^\times,\overline{\mathbb{Q}}_p)\).
A point \(z\) in \(\mathcal{D}(e)(\overline{\mathbb{Q}})p)\) is called classical if there is a classical automorphic eigenform in the corresponding space of overconvergent forms, whose system of Hecke eigenvalues is that defined by \(z\); more generally, recall that points on the eigenvariety corresponds to Hecke packets of overconvergent modular forms.
The main result of this paper is the existence of Hecke packets coming from classical automorphic representations of an inner form of \(\mathrm{GL}(2)\) and two idempotents \(e_1\) and \(e_2\) such that the corresponding point in \(\mathcal{D}(e_1)\) is classical, while the corresponding point in \(\mathcal{D}(e_2)\) is not classical.
The proof of the theorem uses a \(p\)-adic version of a Labesse-Langlands transfer, developped by the author.