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Refined global Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on symplectic groups
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 1 / January 2017
- Published online by Cambridge University Press:
- 19 January 2017, pp. 68-131
- Print publication:
- January 2017
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- Article
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In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg
$L$-functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for 
$\text{Sp}(2n)\times \text{Mp}(2m)$. To support this conjectural identity, we show that when 
$n=m$ and 
$n=m\pm 1$, it can be deduced from the Ichino–Ikeda conjecture in some cases via theta correspondences. As a corollary, the conjectural identity holds when 
$n=m=1$ or when 
$n=2$, 
$m=1$ and the automorphic representation on the bigger group is endoscopic.
