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

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Pith's one-line read The paper establishes, conditional on the refined Gan–Gross–Prasad conjecture, that the Petersson norm ratio for a non-Saito–Kurokawa Siegel cusp form equals an explicit weighted sum of central L-values, with local weights that vanish…

desk verdict First explicit Gan–Gross–Prasad norm formula for non-Saito–Kurokawa degree-two Siegel cusp forms; honest about its conjectural input, but the ramified local computation needs referee-level audit before the result can be fully trusted. read the letter →

arxiv 2608.26007 v1 pith:OWGEZF4T submitted 2026-08-26 math.NT

Biplab Paul , Ameya Pitale , Abhishek Saha , Ralf Schmidt This is my paper · ORCID
classification math.NT MSC 11F4611F3711F7011F66
keywords SiegelcuspformsFourier–JacobiperiodsGan–Gross–PrasadconjecturePeterssonnormshalf-integralweightmodularcentralL-valuesAtkin–Lehnereigenvalues
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to make the refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on $\mathrm{Sp}_4$ explicit enough to compare Petersson norms. For a Hecke eigenform $F$ of degree 2, even weight, and full level that is not a Saito–Kurokawa lift, and for odd squarefree $m$, it derives a conjectural identity expressing $\langle f_m,f_m\rangle/\langle F,F\rangle$ as a finite sum over newforms $f$ of weight $2k-2$ and level dividing $m$, each term being $L(1/2,\pi_F\times\pi_f)/(L(1,\pi_F,\mathrm{Ad})L(1,\pi_f,\mathrm{Ad}))$ times an explicit local weight. The local weights are computed from scratch in the ramified case, and they vanish unless $f$ has Atkin–Lehner eigenvalue 1 at the relevant primes. If the identity is right, it is the first exact norm relation of this kind for non-Saito–Kurokawa forms, and it yields conditional bounds on Petersson norms and Fourier coefficients near the Resnikoff–Saldana shape, together with nonvanishing of central L-values.

What carries the argument

The load-bearing object is the refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on $\mathrm{Sp}_4$: for a cusp form on $\mathrm{Sp}_4$, a genuine cusp form on the metaplectic double cover of $\mathrm{SL}_2$, and a Schwartz function, it predicts that the squared global Fourier–Jacobi period equals $2^{-\beta}\xi(2)^2\xi(4)C_G$ times the ratio of completed L-values and the product of normalized local integrals $\alpha^\#_v$. The paper's new calculation evaluates $\alpha^\#_p$ at the previously open ramified place $v_p(m)=1$. The evaluation proceeds by twisting to the unramified character, passing to the Jacobi-group model, splitting the resulting core integral (87) according to the valuation of the central variable, and using the Cartan decomposition criterion of [19] together with the Macdonald formula for the spherical matrix coefficient of $\pi$. In the special-representation case the endpoint at $s_0=1/2$ is obtained by Bernstein continuation of an absolutely convergent family, with the final rational identity extended to all Satake parameters. The global assembly uses the Shimura–Waldspurger correspondence on the Kohnen plus space to identify each metaplectic representation with a classical newform, so the period becomes $\langle f_m,h\rangle$ and Parseval yields the norm formula.

What would settle it

Take a low-weight non-Saito–Kurokawa Hecke eigenform $F$ (for instance weight 20) and a small odd squarefree $m$; compute $\langle f_m,f_m\rangle/\langle F,F\rangle$ directly from the Fourier expansion and compare it with the right-hand side of formula (4), using known values of the L-functions, so that any mismatch beyond numerical precision refutes the conjectural identity. Computing $\langle f_p,h\rangle$ for a newform $h$ with Atkin–Lehner eigenvalue $-1$ at $p$ would directly test the predicted zero.

Watch

Extended reading notes

Core claim

Let $F\in S_k(\mathrm{Sp}_4(\mathbb{Z}))$ be a Hecke eigenform of even weight $k$ that is not a Saito–Kurokawa lift, and let $f_m$ be the half-integral weight cusp form built from the Fourier coefficients of $F$ by (2). Corollary 3.8 asserts, under the refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods, the identity $$\frac{\langle f_m,f_m\rangle}{\langle F,F\rangle} = \frac{\$pi^{{k+5}}$}{3(2k-3)\Gamma(k)} \sum_{C|m}\sum_{f\in $B^{{\mathrm{new}}$}_{2k-2}(C)} \frac{L(1/2,\pi_F\times\pi_f)}{L(1,\pi_F,\mathrm{Ad})L(1,\pi_f,\mathrm{Ad})}\, r_{f,m,C},$$ where $r_{f,m,C}$ is an explicit product of local factors involving the Satake parameters of $F$ and $f$, and $r_{f,m,C}=0$ whenever the local Atkin–Lehner eigenvalue of $f$ at a prime dividing $C$ is $-1$. The new content is Theorem 2.2, an exact evaluation of the normalized local Fourier–Jacobi integral when the metaplectic representation is an unramified principal series or a special representation and the additive character has conductor with valuation one. Summing the resulting formulas over an orthogonal basis of the Kohnen plus space converts the period identity into the norm formula above.

Load-bearing premise

The entire identity depends on a conjecture rather than a theorem: the squared global Fourier–Jacobi period is exactly the product of certain normalized L-values and local integrals. If that refined Gan–Gross–Prasad prediction is wrong, the norm formula and its analytic consequences do not follow.

Editorial extensions

If this is right

  • For every even weight $k$ and odd squarefree $m$, the Petersson norm ratio for a non-Saito–Kurokawa eigenform is determined by central L-values and local Atkin–Lehner data, with no undetermined constants.
  • Under GRH, $\langle f_m,f_m\rangle \ll_{F,\varepsilon} m^{\varepsilon}$, which is optimal up to the epsilon and is the engine behind the paper's coefficient bounds.
  • For primitive $S$, the conditional bound $|a(F,S)| \ll_{F,\varepsilon} (\min\mathrm{pr}\,S)^{1/2+\varepsilon}(\det S)^{k/2-3/4+\varepsilon}$ improves the previous GRH-conditional bound whenever $\min\mathrm{pr}\,S \le (\det S)^{1/2-\delta}$.
  • If $f_m\neq 0$, at least one central L-value $L(1/2,\pi_F\times\pi_f)$ is nonzero for a newform $f$ of weight $2k-2$ and level dividing $m$; since a positive density of primes have $f_p\neq 0$, those primes certify such a nonvanishing value.
  • The Kohnen–Skoruppa identity for Saito–Kurokawa lifts is recovered as the degenerate case, where the pole of $L(1,\pi_F,\mathrm{Ad})$ leaves a single nonzero term.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: a low-cost numerical test of the identity is to compute $\langle f_p,h\rangle$ for one newform $h$ with Atkin–Lehner eigenvalue $-1$ at $p$; the formula predicts exact vanishing, and a nonzero value would quickly falsify the local weight rule.
  • Editorial inference: the local formula suggests a structural principle for higher-degree Fourier–Jacobi periods: ramified contributions are supported on components whose local Waldspurger lift has Atkin–Lehner eigenvalue 1, so analogous identities for $\mathrm{Sp}_{2n}$ would likely carry the same eigenvalue condition.
  • Editorial inference: because the Fourier–Jacobi framework exists for $\mathrm{Sp}_{2n}$ in every degree, the same strategy would give norm and coefficient bounds for Siegel forms of higher degree once the underlying conjecture is available, a route the Bessel-period method cannot take beyond degree 2.
  • Editorial inference: extending the local computation to non-squarefree $m$ would turn the present squarefree-level formula into a full product formula; the paper leaves this natural next step open.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper computes normalized local Fourier–Jacobi integrals for Sp_4 in the new ramified case v(m)=1, for two types of metaplectic representations: unramified principal series and special representations. The main local theorem, Theorem 2.2, gives explicit formulas for the local quantity alpha^#(pi,sigma;m), including the vanishing condition controlled by the local Atkin–Lehner type. Assuming Xue's refined Gan–Gross–Prasad conjecture (Conjecture 1), the paper uses these local formulas to derive an explicit conjectural identity, Theorem 3.7 and Corollary 3.8, expressing the Petersson norm ratio <f_m,f_m>/<F,F> as a sum over GL_2 newforms of the central value L(1/2, pi_F x pi_f) divided by adjoint L-values, with explicit local weights. Consequences are drawn under GRH: bounds for <f_m,f_m>, bounds of Resnikoff–Saldana shape for Fourier coefficients, and a non-vanishing consequence for central L-values.

Significance. If the local theorem is fully established, the paper provides the first exact norm comparison for degree 2 Siegel cusp forms outside the Saito–Kurokawa case, with a clean Atkin–Lehner condition on the GL_2 side. The reduction chain from the local period to the core integral (87) is explicit and checkable, and the final formula is anchored to several external benchmarks: Xue's unramified normalization alpha^# = 1, the Macdonald formula for spherical matrix coefficients, and standard local L-factors. The paper is honest that the global identity is conditional on Conjecture 1, so that external input is not itself a soundness defect. The conditional applications, especially the GRH-conditional Fourier coefficient bound (171), are concrete and would be useful even if the full refined GGP conjecture remains open. The main internal fragility is the Bernstein continuation at the special-representation endpoint, together with the delegated, non-reproducible computer algebra summation.

major comments (3)
  1. [§2.5, Eqs. (68)–(69)] The passage from the absolutely convergent range 0 <= s0 < 1/2 to the special-representation endpoint s0 = 1/2 is asserted rather than proved. Equations (68) and (69) state that the special-representation integral equals the one-sided limit of a specific four-term combination, although each individual term may have a pole at s0 = 1/2. Uniqueness of the equivariant pairing and the cited lemmas of Xue and Shen do not by themselves show that the four-term combination has a removable singularity, nor that its regular specialization agrees with the endpoint integral defined through the derivative-of-intertwiner inner product (56). If the limit in (69) has a residual pole, then Theorem 2.2(ii), and consequently the p|C local weight 2/(p+1) or 0 in Theorem 1.1, collapses. The authors should supply an explicit pole analysis or a complete Bernstein-continuation argument for the full combination, not merely for the individual scalar family.
  2. [§2.8–§2.11, Eqs. (110), (118), (125), (127)] The evaluation of the core integral is split into 50 cases (10 for I_0, 15 for I_1, 25 for I_2), but full details are supplied only for the first case of each integral (I_{0,1}, I_{1,1}, I_{2,1a}); the remaining cases are described as similar, and the final summation to the compact formula (127) is delegated to Mathematica and ChatGPT with no reproducible script, transcript, or intermediate output. This matters because the final cancellation is needed for both parts of Theorem 2.2, and some displayed intermediate factors, such as (1 - 2q^{-1}) in I_{1,3b}, I_{1,8b}, and I_{2,3b}, are unexplained. The authors should provide a complete case-by-case derivation in an appendix or make available a refereed computer algebra file/transcript that verifies each case and the final simplification.
  3. [§2.11, paragraph after (132)] The rational continuation to exceptional Satake parameters is invoked twice: once for the endpoint limit and once for simplifying the final rational expression when denominators D_{1,sp} or D_{2,sp} vanish. The text states that the right-hand side is understood by simplifying first and then specializing, but it does not prove that the resulting function is independent of the chosen simplification path or that it equals the actual local integral at those parameters. Since Theorem 2.2 is stated for all unramified tempered pi and all special sigma, this missing verification affects the theorem as stated, not merely a generic locus.
minor comments (4)
  1. [Remark 1.3] The consistency check with the Kohnen–Skoruppa identity is described as holding 'up to a constant'; please give the precise constant after simplification, or state explicitly why only the shape of the formula is being compared.
  2. [§2.1, Eq. (20)] The notation Gamma_0(m) and Gamma^0(m) is introduced for ideals and then reused for elements; in (20) the character chi of Gamma is defined only through its action on the upper-left entry, which is fine but should be stated explicitly at that point to avoid ambiguity in the rest of Section 2.
  3. [§3.7, Lemma 3.6] The evaluation of the archimedean integral uses Lemmas 5.1–5.3 of [39] and notes a missing denominator in the statement there; the displayed formula for the matrix coefficient of pi_infty would benefit from a one-line indication of how the normalization is matched to the lowest weight vector, since this factor enters the global constant in Theorem 3.7.
  4. [§1.5] The AI declaration is transparent, but for a paper whose central computation is partly AI-assisted, the exact prompts and versions are not recorded and the statement 'carefully reviewed' is not a substitute for a reproducible computation. Making the computer algebra input and output available as supplementary material would substantially increase confidence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the global identity is honestly conditional on an external conjecture, and the local computation is self-contained.

full rationale

The derivation chain is not circular. The paper's main global identity (4) is explicitly conditional on Conjecture 1 (Xue's refined GGP conjecture, equation (8)), an external conjecture that the paper does not prove or disguise as its own theorem; deriving consequences from an external conjecture is a legitimate conditional derivation, not circularity. The local result Theorem 2.2 is a substantive computation: the normalized local factor alpha^# is defined independently in (32) from a local integral and local L-factors, and the paper evaluates it by explicit integration after reducing to the core integral (87) via the Cartan decomposition and the Macdonald formula. The special-representation endpoint passage in equations (68)-(69) is the only flagged fragility: it relies on an asserted Bernstein continuation cited to Xue [39, Lemma 4.4] and Shen [32, Lemma 5.1], both external to the authors, and the text explicitly warns that the limit is a limit of the complete combination, not of the four separate terms. That is a rigor/correctness concern, not a circular step, because the cited continuation is not equivalent to the target norm identity and is not supplied by fitting. Self-citations, such as [5] for the classification of fixed vectors, [10] for the computational strategy, and [26] for nonvanishing of L(1,pi_F,Ad), are used as background or as references to independently published results; they are not the load-bearing mechanism that makes equation (4) true by construction. No fitted parameter is renamed as a prediction, and the paper does not define a quantity in terms of the ratio it then claims to derive.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data: the formulas depend only on the honest spectral inputs alpha, beta (Satake parameters of pi), delta (Satake parameter of sigma), and the Atkin-Lehner eigenvalues w_p; the central-character parameter gamma = chi_0(vartheta) cancels from the final expressions. The global constant C_G and the Haar measures are conventions fixed explicitly. The main conditional input, Xue's refined GGP conjecture, is listed under axioms, as is the asserted Bernstein continuation argument for the special-representation endpoint. No new particles, forces, or invented mathematical entities are introduced.

assumptions (6)
  • domain assumption Xue's refined Gan-Gross-Prasad conjecture for Fourier-Jacobi periods on Sp_4 (Conjecture 1, equation (8)).
    Load-bearing premise for the entire global identity (4), Theorem 3.7, Corollary 3.8, and all applications in Section 3; it is an unproved conjecture, assumed explicitly by the authors.
  • standard math Macdonald formula for spherical matrix coefficients of GSp_4 (Proposition 2.7) and the Cartan-decomposition criterion of [19, Proposition 3.2] (Lemmas 2.8, 2.10, 2.11, 2.12).
    Standard cited results used to evaluate the matrix coefficient Phi_0 in the core integral (87); treated as background.
  • ad hoc to paper Bernstein continuation for the special-representation family I(Phi_0^m, Phi^J_{sp,s_0}) at s_0 = 1/2 on a Zariski-open locus of Satake parameters, with rational continuation to exceptional parameters.
    Asserted in Section 2.5 (equations (67)-(69)) and Section 2.11; the endpoint value alpha^# = (1 - epsilon_sigma)/(q+1) in Theorem 2.2(ii) depends on the absence of residual or pole contributions at s_0 = 1/2, which is argued by continuation rather than demonstrated directly.
  • standard math Temperedness of pi_F at all primes (Weissauer [37]) and non-vanishing of L(1, pi_F, Ad) (Pitale-Saha-Schmidt [26, Theorem 5.2.1]).
    Used in Section 3.8 to fix beta = 1 in Conjecture 1 and in Remark 1.2; cited from prior literature and not re-proven.
  • standard math Local structure of the Schrodinger-Weil representation, the metaplectic classification (Lemma 2.1 from Berndt-Schmidt [5, Section 5.3]), and the Waldspurger lifting identities (33), (156)-(157).
    Standard framework for the Jacobi group and metaplectic representations; cited from [5], [36], [38], [4].
  • domain assumption Scope restrictions: k even with k at least 4, F of full level Sp_4(Z) and not a Saito-Kurokawa lift; locally, odd residue characteristic, pi unramified tempered with trivial central character, sigma genuine unitary tempered and not an even Weil representation.
    These assumptions define the domain of the theorem; they are stated openly and do not force the target result.

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Pith. "Pith review of An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms." pith.science (2026). https://pith.science/paper/OWGEZF4T

@misc{pith2026260826007,
  author       = {Pith},
  title        = {Pith review of: An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OWGEZF4T}},
  note         = {Machine review of arXiv:2608.26007}
}
abstract

We compute the local integrals appearing in the refined Gan--Gross--Prasad conjecture for Fourier--Jacobi periods of $\mathrm{Sp}_4$ in new ramified cases and use this to formulate an explicit conjectural identity relating Petersson norms of degree 2 Siegel cusp forms and associated half-integral weight forms. We note consequences of our identity for the growth of Petersson norms, the size of Fourier coefficients, and non-vanishing of central $L$-values.

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    Fourier-Jacobi periods of classical Saito-Kurokawa lifts.Ramanujan J., 45(1):111–139, 2018

    Hang Xue. Fourier-Jacobi periods of classical Saito-Kurokawa lifts.Ramanujan J., 45(1):111–139, 2018. FOURIER–JACOBI PERIODS 53 Department of Mathematics, IIT Bhubaneswar, Argul, Khordha, Odisha 752051, India Email address:bpaul@iitbbs.ac.in Department of Mathematics, Universi...

    2018

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.