#### Vol. 295, No. 1, 2018

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Distinguished residual spectrum for $\mathop{\rm GL}_2(D)$

### Mahendra Kumar Verma

Vol. 295 (2018), No. 1, 241–256
##### Abstract

Let $G={GL}_{2}\left(D\right)$ where $D$ is a quaternion division algebra over a number field $F$ and $H={Sp}_{2}\left(D\right)$ is the unique inner form of ${Sp}_{4}\left(F\right)$. We study the period of an automorphic form on $G\left(\mathbb{A}\right)$ relative to $H\left(\mathbb{A}\right)$ and we provide a formula, similar to the split case, for an automorphic form in the residual spectrum. We confirm the conjecture due to Dipendra Prasad for noncuspidal automorphic representations, which says that symplectic period is preserved under the global Jacquet–Langlands correspondence.

##### Keywords
symplectic period, Jacquet–Langlands correspondence
##### Mathematical Subject Classification 2010
Primary: 11F41, 11F67, 11F70