Resnikoff–Saldaña bound for Fourier coefficients of degree-2 Siegel cusp forms

Prove the Resnikoff–Saldaña conjectural bound that, for a degree-2 Siegel Hecke cusp form F and a fundamental Fourier-index matrix S, the Fourier coefficient a(F,S) satisfies |a(F,S)| \ll_{F,\epsilon} (\det S)^{k/2-3/4+\epsilon}.

Background

The paper studies Fourier coefficients a(F,S) of degree-2 Siegel cusp forms through Fourier–Jacobi coefficients and half-integral-weight modular forms. The authors obtain a conditional bound involving the smallest integer represented by S, assuming the refined Gan–Gross–Prasad conjecture and GRH.

They compare this result with the stronger conjectural estimate of Resnikoff and Saldaña, which removes the dependence on the smallest represented integer and has the expected exponent k/2−3/4. The paper explicitly describes this conjecture as famously difficult and does not prove it.

References

A famously difficult conjecture of Resnikoff and Saldana predicts that \begin{equation}\label{e:RSboundintro}|a(F,S)| \ll_{F,\epsilon} (\det S){\frac{k}2 -\frac{3}4 + \epsilon}.\end{equation}

e:RSboundintro:

∣a(F,S)∣≪F,ϵ(det⁡S)k2−34+ϵ.|a(F,S)| \ll_{F,\epsilon} (\det S)^{\frac{k}2 -\frac{3}4 + \epsilon}.

— An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms  (2608.26007 - Paul et al., 26 Aug 2026) in Section 1, subsection “The global identity and some consequences,” and Section 3, subsection “Upper bounds on Petersson inner products and Fourier coefficients”