Let F be a degree-2 Siegel cuspidal Hecke eigenform, with associated automorphic representation pi_F of GSp_4( A), and let g be an associated half-integral-weight form, with automorphic representation sigma_g. For compatible Fourier--Jacobi data, the conjecture predicts an explicit identity of the form frac| P_FJ(F,g)|^2langle F,F rangle,langle g,g rangle=C_∞(F,g),fracL!(tfrac12,pi_F×sigma_g)L!(1,pi_F,operatornameAd)L!(1,sigma_g,operatornameAd)∏_v alpha_v(F,g), where P_FJ(F,g) is the Fourier--Jacobi period, C_∞(F,g) is the specified archimedean factor, and alpha_v(F,g) are the normalized local factors. The conjecture asserts this exact global relation, including the Petersson norms, adjoint factors, central L-value, and local terms.

Refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods of degree-2 Siegel cusp forms

Let $F$ be a degree- $2$ Siegel cuspidal Hecke eigenform, with associated automorphic representation $\pi_F$ of $\mathrm{GSp}_4(\mathbb A)$ , and let $g$ be an associated half-integral-weight form, with automorphic representation $\sigma_g$ . For compatible Fourier--Jacobi data, the conjecture predicts an explicit identity of the form $\frac{|\mathcal P_{\mathrm{FJ}}(F,g)|^2}{\langle F,F\rangle\,\langle g,g\rangle}=C_\infty(F,g)\,\frac{L\!\left(\tfrac12,\pi_F\times\sigma_g\right)}{L\!\left(1,\pi_F,\operatorname{Ad}\right)L\!\left(1,\sigma_g,\operatorname{Ad}\right)}\prod_v\alpha_v(F,g)$ , where $\mathcal P_{\mathrm{FJ}}(F,g)$ is the Fourier--Jacobi period, $C_\infty(F,g)$ is the specified archimedean factor, and $\alpha_v(F,g)$ are the normalized local factors. The conjecture asserts this exact global relation, including the Petersson norms, adjoint factors, central $L$ -value, and local terms.

References

Primary source

arXiv

Additional references

An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

An August 2026 paper adds important local calculations and sharper tests, but the conjectured global identity remains unproved.

The conjecture predicts an exact formula linking Fourier–Jacobi periods of degree- $2$ Siegel cusp forms with central $L$ -values, adjoint factors, norms, and local terms. Hang Xue formulated the broader identity and established special cases, but not the full degree- $2$ case.

Known results

Hang Xue proved the identity is well-defined and independent of the auxiliary set of places $S$ .
Hang Xue established cases with $n=m=1$ .
Hang Xue established the $n=2$ , $m=1$ endoscopic case; these do not settle the full conjecture.

August 26, 2026 local calculations

A new paper computes local Fourier–Jacobi integrals for $\mathrm{Sp}_4$ in additional ramified cases and formulates an explicit conditional identity for degree- $2$ Siegel cusp forms and associated half-integral-weight forms. It derives consequences for Fourier coefficients, Petersson-norm growth, and central $L$ -value nonvanishing, but does not prove the global conjecture.

Current status (as of September 2026): Special cases and new local calculations are established, while the full refined global identity remains open and the latest advance is unverified.

Sources

No solutions have been posted yet.