Ramanujan Bounds for Siegel Cusp-Form Coefficients

Definitions and mathematical statement

Let $F$ be a holomorphic Siegel Hecke cusp form of degree two and weight $k$ for $\operatorname{Sp}_4({\mathbb{Z}})$, as in the supplied statement. Write \[ \mathfrak H_2=\{Z\in M_2({\mathbb{C}}):Z^{\mathsf T}=Z,\ \operatorname{Im}Z>0\}. \] Use the conventional Fourier-index lattice \[ \Lambda=\left\{S=\begin{pmatrix}a&b/2\\b/2&c\end{pmatrix}:a,b,c\in{\mathbb{Z}},\ a>0,\ 4ac-b^2>0\right\}. \] Thus $S$ is positive definite and half-integral. Its associated integral binary quadratic form is $ax^2+bxy+cy^2$, with discriminant \[ D(S)=b^2-4ac=-4\det S. \] Let $\Lambda_0\subseteq\Lambda$ consist of the indices for which $D(S)$ is a negative fundamental discriminant, meaning the discriminant of an imaginary quadratic field. These definitions reconstruct the conventional meaning of the source's notation; the supplied excerpt does not explicitly define $\Lambda$ or $\Lambda_0$. Define the coefficients by \[ F(Z)=\sum_{S\in\Lambda}a(F,S)\exp\!\left(2\pi i\operatorname{tr}(SZ)\right),\qquad Z\in\mathfrak H_2. \]

The first requested assertion is that, for each fixed $F$ as above, there exist $\delta>0$ and $C_F>0$, independent of $S$, such that \[ |a(F,S)|\le C_F(\det S)^{(k-1)/2-\delta}\qquad\text{for every }S\in\Lambda_0. \] The source calls $\delta$ fixed but does not specify whether it must also be independent of $F$ and $k$. The displayed formulation therefore records the fixed-$F$ assertion and leaves the stronger uniform interpretation unresolved. The source explicitly permits a conditional treatment using standard conjectures on $L$-functions, including the generalized Riemann hypothesis (GRH), and expected period identities. Any such dependencies must be stated.

The stronger conjectural target, attributed in the source to Reznikov--Saldana, is \[ \forall\epsilon>0\ \exists C_{F,\epsilon}>0\ \forall S\in\Lambda_0:\qquad |a(F,S)|\le C_{F,\epsilon}(\det S)^{k/2-3/4+\epsilon}. \] The source additionally proposes this stronger estimate for every $S\in\Lambda$, provided that $F$ is not a Saito--Kurokawa lift. That exclusion is supplied for the extension to $\Lambda$; the supplied fundamental-discriminant formulation does not impose it.

For context, the supplied excerpt attributes to Kohnen the unconditional estimate \[ |a(F,S)|\ll_{F,\epsilon}(\det S)^{k/2-13/36+\epsilon}\qquad(S\in\Lambda_0,\ \epsilon>0). \] Its description as the current best unconditional bound is a source claim, not an independently established literature assessment here. The excerpt also attributes a conditional estimate under GRH to Jaasaari--Lester--Saha. The apparent reading of its damaged typesetting is \[ |a(F,S)|\ll_{F,\epsilon} \frac{(\det S)^{k/2-1/2}}{(\log(\det S))^{1/8}}. \] This transcription requires checking. In particular, although the excerpt says that the estimate holds for all $S\in\Lambda_0$, the displayed logarithmic denominator does not define a positive real expression when $\det S\le1$; no small-determinant convention is supplied. It is therefore recorded only as an apparent large-determinant reading, not as a precise all-index theorem.

The excerpt motivates these estimates by an expected period identity following from the Gan--Gross--Prasad conjectures. It involves, for a class-group character $\chi$, a sum of the form \[ \sum_{\substack{[S]\in\operatorname{SL}_2({\mathbb{Z}})\backslash\Lambda_0\\D(S)=d}} \chi([S])a(F,S) \] and the central value $L(1/2,F\times\theta_\chi)$, where the source does not define the normalization of $\theta_\chi$. The extracted formula contains a power resembling $d^{k-1}$, but its placement of a square, its use of signed or absolute discriminant, and its proportionality factors are unclear. No precise period identity is asserted in this draft.

Short English statement

For a fixed holomorphic Siegel Hecke cusp form of degree two, seek a power saving in its Fourier coefficients at positive definite indices of fundamental discriminant. The stronger target is the supplied Reznikov--Saldana estimate, with a proposed extension to all positive definite indices after excluding Saito--Kurokawa lifts. Conditional arguments are explicitly permitted. Uniformity of the saving, damaged background formulas, and the supplied literature assessment require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005530$, title Ramanujan Bounds for Siegel Cusp-Form Coefficients, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is 2023.
• Supplied report citation: Group Actions and Harmonic Analysis in Number Theory (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/22, pp. 1195--1262. https://doi.org/10.4171/owr/2023/22. The excerpt identifies the primary request as Question 3, the Kohnen estimate as Theorem 6, the conditional Jaasaari--Lester--Saha estimate as Theorem 7, and the Reznikov--Saldana target as Conjecture 2.
• Supplied literature source labeled "Blomer--Pohl theorem": https://arxiv.org/abs/1402.4635.
• Supplied literature source labeled "published sup-norm paper": https://doi.org/10.1353/ajm.2016.0032.
• Literature assessment checked at the exact supplied date 2026-08-22: the registry labels the problem partially_solved and states that power-saving sup-norm bounds hold on each fixed compact set for degree-two Hecke--Maass forms, while a stronger global uniform bound remains unresolved. This is supplied dated curation, not independently verified. Its subject differs from the holomorphic Fourier-coefficient problem formulated here, so it does not establish partial resolution of this statement.
• The registry labels the statement corrected_verified and supplies the assertion that it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim; no independent source consultation or mathematical verification is claimed here. The background explicitly requests independent verification of time-sensitive status and cited claims.