An arXiv preprint reports explicit formulas for local Fourier-Jacobi integrals in new ramified cases for Sp4 and uses them to formulate a conjectural link between degree-2 Siegel cusp forms and associated half-integral-weight forms. The local results are stated under specific representation-theoretic assumptions. The global identity, however, is explicitly conditional on the refined Gan-Gross-Prasad conjecture, so the paper does not claim to have proved that part.
The local theorem fixes the Fourier-Jacobi parameter at normalized valuation one and treats an unramified metaplectic principal series alongside a special representation. At the global level, it studies a full-level Hecke eigenform of even weight that is not a Saito-Kurokawa lift, together with a positive odd squarefree index and its associated half-integral-weight form. The local proof combines twisting, a Jacobi-group reformulation, a damping factor for convergence, valuation splitting, Cartan decomposition, the Macdonald formula and local L-factor normalization.
Two exact local cases
In the principal-series case, the normalized local factor is given by a closed rational expression in the Satake parameters and the value of the defining character at a local uniformizer. Put simply, the paper replaces the integral with an explicit formula in the local data. The result applies to the unramified principal-series setting at valuation one.
The special-representation case has a sharper split. The normalized factor is zero when the associated local sign is positive, and it equals 2 divided by one plus the residue-field size when that sign is negative. This second formula is also restricted to the stated valuation-one assumptions.
From local factors to a global formula
Under the refined GGP conjecture, a weighted sum of squared Petersson pairings between the associated half-integral-weight form and an orthogonal basis is expressed through the Petersson norm of the Siegel form, a central Rankin-Selberg L-value, two adjoint L-values and a product of local factors. The statement connects the pairings with global L-values and local information at primes dividing the index. That global identity remains conjectural rather than becoming a theorem in the paper.
Orthogonality gives a second version of the proposed result: an explicit formula for the ratio of the Petersson norm of the half-integral-weight form to that of the Siegel form. Instead of summing over individual basis elements, this expression sums over associated GL2 representations and combines local weights with a central-to-adjoint L-value ratio. The norm identity is presented under the same refined GGP assumption.
The consequences stay conditional
The paper also states several consequences only after adding further assumptions. With both the refined GGP conjecture and GRH, it derives an essentially optimal conditional upper bound for the Petersson norm of the associated half-integral-weight form: the norm is bounded, up to a constant depending on the fixed Siegel form and an arbitrarily small positive parameter, by the index raised to that parameter. The same assumptions produce a conditional bound for a Fourier coefficient, involving the smallest odd prime represented by its index matrix and the matrix determinant.
Another conditional implication concerns non-vanishing. Under the refined GGP conjecture, if the associated half-integral-weight form is nonzero, then at least one associated classical newform has a nonzero central Rankin-Selberg L-value. The statement therefore starts from non-vanishing of the half-integral-weight form rather than proving it in every case.
What the calculation does not settle
The local calculation has clear boundaries: it covers only valuation-one instances of the specified principal-series and special-representation cases, while the global setup is full-level and excludes Saito-Kurokawa lifts. The paper does not turn the global statement into an unconditional result. It reports that the local core-integral calculations and simplifications were formalized and machine-verified in Lean 4, with a public development and registration in the Palomar Registry.
The document is identified as arXiv:2608.26007v2 and dated 21 September 2026.
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