The arithmetic Gan–Gross–Prasad conjecture for fixed level

Let $K\subset \widetilde{H G}({\mathbb A}_f)$ be an open compact subgroup. Let $\pi$ be an automorphic representation with trivial restriction to $Z^{\mathbb Q}({\mathbb A})$ and lying in a cohomological tempered Arthur packet. Let ${\mathcal Z}_{K,0}$ be the cyclic Hecke submodule generated by the distinguished cohomologically trivial diagonal cycle, and let $R$ be the tensor-product representation defining $L(s,\pi,R)$.

Arithmetic Gan–Gross–Prasad conjecture. Conditions (a) $\dim\operatorname{Hom}_{{\mathscr H}_K}(\pi_f^K,{\mathcal Z}_{K,0})=1$ and (b) $\operatorname{ord}_{s=1/2}L(s,\pi,R)=1$, together with the one-dimensionality and nonvanishing condition for $\operatorname{Hom}_{\widetilde H({\mathbb A}_f)}(\pi_f,\mathbb C)$, are equivalent and imply (c) $\operatorname{Hom}_{{\mathscr H}_K}(\pi_f^K,{\mathrm{Ch}}^{n-1}(M_K(\widetilde{H G}))_{\mathbb C,0})\neq0$. If $E=F$, these conditions are also equivalent to the strengthened condition that the latter Hom-space has dimension one and satisfies the same period nonvanishing condition.

This is a fixed-level arithmetic refinement of Gan–Gross–Prasad relating diagonal cycles, central $L$-function vanishing, and automorphic periods. The source presents it as conjectural and notes that it is based on widely open standard conjectures.