The global Gan-Gross-Prasad (GGP) conjecture [W. T. Gan et al., Astérisque 346, 1–109 (2012;Zbl 1280.22019)] relates the non-vanishing of certain global period integrals of automorphic forms to the non-vanishing of central values of \(L\)-functions of Rankin-Selberg type. The conjecture is also expected to have a refined form, which gives a precise formula between the global periods and central \(L\)-values. Such formulae, asJ. L. Waldspurger’s formula [Compos. Math. 54, 173–242 (1985;Zbl 0567.10021)] for instance, could have various applications in the study of arithmetic problems.
For the cases of Bessel periods, the refined conjecture has been formulated byA. Ichino andT. Ikeda [Geom. Funct. Anal. 19, No. 5, 1378–1425 (2010;Zbl 1216.11057)] for the pair \((\mathrm{SO}_n,\mathrm{SO}_{n+1})\), byN. Harris [Int. Math. Res. Not. 2014, No. 2, 303–389 (2014;Zbl 1322.11047)] for the pair \((\mathrm{U}_n, \mathrm{U}_{n+1})\), and byY. Liu [J. Reine Angew. Math. 717, 133–194 (2016;Zbl 1404.11065)] for general Bessel periods for classical groups. The refined conjecture for Fourier-Jacobi periods was only considered for the pair \((\mathrm{U}_n, \mathrm{U}_n)\) before by the author of this paper [Adv. Math., 262, 1130–1191 (2014;Zbl 1301.11051)].
In the first part of this paper, the author propose a conjectural formula for Fourier-Jacobi periods in symplectic-metaplectic cases. Let \(F\) be a number field with ring of adeles \(\mathbb{A}\), and \(\psi: F\backslash \mathbb{A}\longrightarrow \mathbb{C}^\times\) be a fixed non-trivial additive character. For two symplectic spaces \(W\supset W_0\) of codimension \(2r\) over \(F\), denote \(G=\mathrm{Sp}(W)\), \(G_0=\mathrm{Sp}(W_0)\), and \(\widetilde G_0=\mathrm{Mp}(W_0)\), the metaplectic double cover of \(\mathrm{Sp}(W_0)\). Let \(\pi\) be an irreducible cuspidal tempered automorphic representation of \(G(\mathbb{A})\), and \(\pi_0\) be an irreducible cuspidal tempered genuine automorphic representation of \(\widetilde G_0(\mathbb{A})\). For automorphic forms \(\varphi\in \pi\) and \(\varphi_0\in \pi_0\), one defines the Fourier-Jacobi period\[\mathcal{FJ}_{\psi}(\varphi,\varphi_0, \phi)=\int_{[G_0]}\int_{[\mathcal{H}_{W_0}]}\int_{[N_{r-1}]}\varphi_0(g_0)\varphi(uhg_0)\overline{\psi_{r-1}(u)\theta_\psi^\phi(h\cdot g_0)}\,du\,dh\,dg_0,\]where \(\mathcal{H}_{W_0}=W_0\ltimes F\) is the Heisenberg group attached to \(W_0\); \(\theta_\psi^\phi\) is a theta series on \(\mathcal{H}_{W_0}(\mathbb{A})\rtimes \widetilde G_0(\mathbb{A})\) with respect to \(\psi\) with \(\phi\in \mathcal{S}(\mathbb{A}^n)\) being a Schwartz function; \(N_{r-1}\) is the unipotent radical of the parabolic subgroup \(P_{r-1}\) in \(G\) whose Levi subgroup is isomorphic to \(\mathrm{GL}_1^{r-1}\times\mathrm{Sp}(W_0+\mathbb{H})\) (here \(\mathbb{H}\) is a hyperbolic plane over \(F\)), and \(\psi_{r-1}\) is an automorphic character on \(N_{r-1}(\mathbb{A})\) which is stable under the conjugation action of \(\mathcal{H}_{W_0}(\mathbb{A})\rtimes G_0(\mathbb{A})\) (here \(\mathcal{H}_{W_0}\) embeds into \(N_{r-1}\) as a unipotent subgroup). The convergence of \(\mathcal{FJ}_{\psi}(\varphi,\varphi_0, \phi)\) is guaranteed by the cuspidality of \(\pi\) and \(\pi_0\).
On the other hand, one can define the (partial) Rankin-Selberg \(L\)-function \(L_\psi^S(s,\pi\times \pi_0)\) for a sufficient large finite set of places \(S\), which depends on the additive character \(\psi\). Suppose that \(\varphi=\otimes_v \varphi_v\) and \(\varphi_0=\otimes_v \varphi_{0,v}\) are decomposable, the author proposed the conjectural formula:\[|\mathcal{FJ}_{\psi}(\varphi,\varphi_0, \phi)|^2=\frac{\Delta_G^S}{|S_{\pi}|\cdot |S_{\pi_0}|}\frac{L^S_\psi(\frac{1}{2},\pi\times \pi_0)}{L^S(1,\pi,\mathrm{Ad})L_\psi^S(1,\pi_0,\mathrm{Ad})}\cdot \prod_{v\in S} \alpha_v(\varphi_v, \varphi_{0,v},\phi_v),\]where \(\Delta_G^S=\prod_{i=1}^{n+r} \zeta_F^S(2i)\) with \(n=\frac{\dim W_0}{2}\); \(S_\pi\) and \(S_{\pi_0}\) are centralizers of global \(L\)-parameters of \(\pi\) and \(\pi_0\), respectively; \(L^S(1,\pi,\mathrm{Ad})\) and \(L_\psi^S(1,\pi_0,\mathrm{Ad})\) are adjoint \(L\)-functions of \(\pi\) and \(\pi_0\) respectively (where the later one depends also on the additive character \(\psi\)); and \(\alpha_v(\varphi_v, \varphi_{0,v}, \phi_v)\) is a local linear form defined by integration of matrix coefficients, whose convergence is given in the Section 2 of this paper. In Section 4 of this paper, the author gives detailed calculations for the local linear form \(\alpha_v(\varphi_v, \varphi_{0,v}, \phi_v)\) in unramified situation, which provides foundation to the above conjectural formula. Note that the identity also relies on precise choices of measures in the Fourier-Jacobi period \(\mathcal{FJ}_{\psi}(\varphi,\varphi_0, \phi)\).
In the second part of this paper, the author studies the compatibility of the conjectural formula with that proposed by Ichino-Ikada for special orthogonal groups via theta correspondence. In particular, based on the known cases on the orthogonal side, the author verifies the conjectural formula for the pair \((\mathrm{SL}_2, \mathrm{Mp}_2)\), and also for the pair \((\mathrm{Sp}_4, \mathrm{Mp}_2)\) when the automorphic representation on the bigger group is endoscopic.
The paper is well written with adequate details. It should be a standard reference for relevant works.

Reviewer: Bin Xu (Chengdu)

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.