A \(p\)-adic Labesse-Langlands transfer (Q2408123)

The author studies a Langlands-Labesse transfer of $p$-adic automorphic forms from the group $\tilde{G}$ of units in a definite quaternion algebra to its subgroup $G$ of norm 1 elements. This is an application of the method first used by \textit{G. Chenevier} [Duke Math. J. 126, No. 1, 161--194 (2005; Zbl 1070.11016)], who studied the $p$-adic Jacquet-Langlands transfer between $\tilde{G}$ and $\mathrm{GL}_2$. \par The method is to build eigenvarieties $\tilde{\mathcal{D}}$ and $\mathcal{D}$ respectively for $\tilde{G}$ and $G$ whose points correspond to a family of $p$-adic automorphic forms, and a morphism $\zeta:\tilde{\mathcal{D}}\rightarrow \mathcal{D}$. This morphism is compatible with the associated weight data and Hecke-algebra data. Also, since classical modular forms for $\tilde{G}$ and $G$ respectively occupy a dense subset in $\tilde{\mathcal{D}}$ and $\mathcal{D}$, the author also proved that $\zeta$ interpolates the classical Langlands-Labesse transfer.