An arXiv preprint sets out an explicit mathematical link between degree-2 Siegel cusp forms and central L-values, the values of associated L-functions at their central point. The link is expressed through Fourier–Jacobi periods, special integrals attached to the forms, and Petersson pairings. The paper also traces consequences for Petersson norms and Fourier coefficients. But the main global identity is obtained only after assuming the refined Gan–Gross–Prasad conjecture. The work therefore presents a concrete formula and conditional consequences within a conjectural framework, rather than an unconditional result.
This is a theoretical study, not an experiment. It uses no empirical sample; it analyzes specified mathematical forms and their Fourier coefficients. Its evidence consists of mathematical identities, calculations and conditional implications rather than observations from a study population.
An explicit local calculation
At the local level, the paper splits the central variable by valuation into three pieces, I0, I1 and I2. It evaluates these pieces using Cartan-decomposition criteria, Macdonald’s formula for spherical matrix coefficients, geometric-series summation and local L-factors. The procedure yields normalized local Fourier–Jacobi factors for the ramified cases.
For valuation-one local data, the normalized factor has an explicit formula in the principal-series case. In the special-representation case, it is 0 when εσ = 1 and 2/(q + 1) when εσ = −1. The calculation is limited to the local representation-theoretic cases covered by the preprint.
To build the global statement, the paper combines a classical interpretation of the adelic period with archimedean calculations, the Shimura–Waldspurger correspondence, a Hecke-basis summation and Parseval’s identity. This chain connects the local factors to global expressions involving Petersson pairings, norms and central L-values.
The main consequences remain conditional
Assuming Conjecture 1, the global theorem says that the sum of squared Petersson pairings over the constructed basis equals a global central L-value ratio multiplied by local factors. A related corollary expresses the Petersson norm ratio of the associated half-integral-weight form as a sum over relevant automorphic representations of central L-value ratios weighted by local factors.
Those upper-bound results require GRH as well as the refined Gan–Gross–Prasad conjecture. Under both assumptions, the paper derives ⟨f_m, f_m⟩ ≪_{F,ε} m^ε. It also bounds |⟨f_m, h⟩| by an m^{-1/2}(km)^ε factor times square-root Petersson norms and a k-dependent gamma factor.
Another conditional estimate concerns Fourier coefficients. In the stated fundamental-discriminant setting, the paper gives, under GRH and Conjecture 1, |a(F,S)| ≪_{F,ε} (min pr S)^{1/2+ε}(det S)^{k/2−3/4+ε}. The bound involves the smallest represented odd prime and the determinant, and is limited to that stated setting.
A further implication for non-vanishing
The non-vanishing implication is conditional too. For each positive odd squarefree m for which the associated Fourier–Jacobi form is nonzero, Conjecture 1 implies the existence of a newform of the stated weight and level dividing m with a nonzero central L-value. The result starts only when the relevant Fourier–Jacobi form is already known to be nonzero.
The introduction also states that infinitely many qualifying m exist and that a positive density of odd primes has the required non-vanishing property. That context does not remove the main qualification: the associated L-value conclusion remains conditional.
Taken together, the paper offers explicit local calculations and a global formula with conditional consequences; it does not establish the refined Gan–Gross–Prasad conjecture or make the strongest estimates unconditional. The stated results remain tied to the representation-theoretic cases and mathematical forms covered by the preprint. The document identifies itself as an arXiv version-1 preprint.
Paper data and sources
Original title: An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms
Authors: Biplab Paul, Ameya Pitale, Abhishek Saha, Ralf Schmidt
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text