Poisson summation conjecture for the triple product space

Let $F$ be the number field, let $X$ and $Y$ be the spaces in the paper, and let $X^0(F)$ and $Y^0(F)$ denote their open $F$-rational loci. For every place $v$ of $F$, choose local Schwartz spaces $\mathcal{S}(X(F_v) \times Y(F_v),\mathcal{L}_{\psi_v})$, and let $\mathcal{S}(X(\mathbb{A}_F) \times Y(\mathbb{A}_F),\mathcal{L}_{\psi})$ be the associated adelic Schwartz space. Let $\mathcal{F}$ be the Fourier transform and let (Sup) denote the support condition described in the paper.

Poisson summation conjecture. One can choose these Schwartz spaces so that the properties (FT1), (FT2), (FT3), (basic:norm), (rapidlydecreasing), (Frechet), (nA:ratio), (Arch:ratio), and (zeta:basic) hold, and, for every $f \in \mathcal{S}(X(\mathbb{A}_F) \times Y(\mathbb{A}_F),\mathcal{L}_{\psi})$ satisfying (Sup),

$$\sum_{(x,y) \in X^\circ(F) \times Y^\circ(F)}\mathrm{ev}_{(x,y)}(f)=\sum_{(x,y) \in X^\circ(F) \times Y^\circ(F)}\mathrm{ev}_{(x,y)}(\mathcal{F}(f)).$$

This is the most optimistic Poisson summation statement needed in the paper. Its heuristic support comes from defining the local Schwartz spaces through a suitable Plancherel formula and the expected analytic properties of Langlands $L$-functions; establishing it remains conditional on those expected properties.