Refined Gan–Gross–Prasad identity for Fourier–Jacobi periods

Establish the refined Gan–Gross–Prasad identity for Fourier–Jacobi periods on the symplectic group GSp4, relating the squared global Fourier–Jacobi period to the relevant completed central L-value, adjoint L-values, the Tamagawa-measure normalization, and the product of normalized local Fourier–Jacobi integrals for tempered automorphic representations that are generic almost everywhere.

Background

The paper’s principal global identity for Petersson norms of degree-2 Siegel cusp forms is conditional on the refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on GSp4. The conjecture concerns a cuspidal automorphic representation π of GSp4 and a genuine cuspidal automorphic representation σ of the metaplectic double cover of GL2, together with a Schwartz function and factorizable automorphic vectors.

It predicts an exact factorization of the squared global Fourier–Jacobi period into a global central L-value ratio and normalized local factors. The authors compute new non-archimedean local factors and use the conjectural identity to derive their global Petersson-norm formula, bounds, and non-vanishing consequences; however, the global identity itself remains conjectural in the paper.