Global Gan–Gross–Prasad Conjecture (Fixed-Representation Form)

Global Gan–Gross–Prasad Conjecture (Fixed-Representation Form)

Definitions and mathematical statement

For the relevant classical-group pair, let $H$ be the subgroup in Gan--Gross--Prasad, $\nu=\bigotimes'_v\nu_v$ its Bessel or Fourier--Jacobi data, and $R$ the prescribed representation defining the finite $L$-function. For each fixed tempered cuspidal automorphic representation $\pi$ occurring with multiplicity one, the assertion is \[ \mathcal F(\nu)|_\pi\ne0 \quad\Longleftrightarrow\quad \left(\forall v,\ \operatorname{Hom}_{H(F_v)}(\pi_v,\nu_v)\ne0\right) \ \text{and}\ L(\pi,R,\tfrac12)\ne0. \] The left side says that the source-defined global period is not identically zero on $\pi$. The local Hom-spaces test distinction at every place. The form signs, covers, auxiliary characters, and inverse twist in $R$ are retained from Sections 23--24. Neither a packet-existence statement nor a refined numerical period formula is substituted.

Formulation and concept references: [S1].

Short English statement

For each allowed automorphic representation, is its global period nonzero exactly when every local period is allowed and the central $L$-value is nonzero?

Research attempt

Source audit and current frontier.

Sections 23--24 of Gan--Gross--Prasad were inspected, retaining the conjugate Weil representation in the Fourier--Jacobi pairing and the $\mu^{-1}$ twist in $R$. Conjecture 24.1 is for the fixed representation, not merely existence somewhere in a packet. Its split corank-one Rankin--Selberg remarks are consistency checks, not substitutes for all allowed cases. [S1] Boisseau--Lu--Xue's 2026 manuscript, the third in their unitary Fourier--Jacobi series, supplies a global nonvanishing statement, local detection and a refined formula in Theorems 1.1, 1.3 and 1.4. This is substantial specific progress, not a result for all orthogonal and symplectic/metaplectic cases in the catalogue. Those are the cited manuscript's theorems; its full proofs have not independently been verified here. [E1]

Attack route.

Local multiplicity one suggests factorizing the global period and determining its scalar by a relative trace formula. A support comparison, identifying only whether the scalar vanishes, would already suffice. Alternatively one might globalize local nonzero test vectors. We show exactly why the latter stops at a scalar, and prove a spectral-filter identity exposing the tail estimate required to isolate one representation rather than a packet sum.

Attempt: local uniqueness leaves a global scalar.

Let $E_v$ be the algebraic smooth local test space: $\pi_v$ in the Bessel case, or $\pi_v\otimes\overline{\nu_v}$ when the Fourier--Jacobi pairing is packaged as an invariant linear form. This notation retains the source's dual conventions. Assume for the following lemma that the relevant invariant form spaces have dimension one, with nonzero forms $\ell_v$, and that reference vectors can be normalized by $\ell_v(e_v)=1$, compatibly with the restricted tensor product outside finitely many places. These local and unramified hypotheses are not proved anew for all catalogue cases. [S1]

For any invariant linear form $P$ on $E=\bigotimes'_vE_v$, \begin{equation}\label{wsref-0-33-30-30-30-36-39-37-37-3a-73-63-61-6c-61-72} P(\otimes_v f_v)=c\prod_v\ell_v(f_v),\qquad c=P(\otimes_v e_v). \end{equation} The product is finite, since $f_v=e_v$ almost everywhere. Freeze all but one factor: invariance makes the resulting linear form a multiple of $\ell_v$, and evaluation at $e_v$ determines the multiple. Induction over the finitely many modified factors proves (30006977:scalar); linearity extends it to the algebraic tensor product. If a local form space is zero, the same argument kills every pure tensor. On completed smooth spaces, continuity and density are needed to extend that conclusion. Convergence and continuity of the actual cuspidal period must remain part of the source-defined construction. [S1]

Thus a nonzero global period forces the local conditions. Conversely all local conditions construct an abstract nonzero tensor functional, but do not show that the automorphic period has $c\ne0$. The zero form satisfies the same invariance identities. Local uniqueness therefore cannot supply arithmetic nonvanishing. This argument also does not prove the necessary central-$L$ condition.

An explicit spectral-localization requirement.

Let $e_0,e_1,\ldots$ be orthonormal eigenvectors of a commuting family of diagonal normal operators, and let $b_j\ge0$ be spectral weights. Suppose for each $1\le i\le M$ an operator $T_i$ has distinct eigenvalues $t_i(0),t_i(i)$. Define \[ Q_M=\prod_{i=1}^M\frac{T_i-t_i(i)I}{t_i(0)-t_i(i)}, \qquad Q_Me_j=q_M(j)e_j. \] Then $q_M(0)=1$ and $q_M(i)=0$ for $1\le i\le M$. Consequently, whenever the series converges, \begin{equation}\label{wsref-0-33-30-30-30-36-39-37-37-3a-66-69-6c-74-65-72} \sum_{j\ge0}b_j|q_M(j)|^2 =b_0+\sum_{j>M}b_j|q_M(j)|^2. \end{equation} A comparison calculation with a positive lower limit on the left side and \[ \sum_{j>M}b_j|q_M(j)|^2\longrightarrow0 \] would force $b_0>0$. In a correctly regularized relative trace formula the intended $b_0$ detects the period of the fixed representation, and the comparison side would detect its central $L$-value. This is an explicit attack requirement, not a trace formula claimed for the unsolved cases. Isolation and regularization are substantive steps of the contemporary unitary argument. [E1]

Finite interpolation alone does not prove the tail condition. A factor separating eigenvalues $0$ and $\epsilon$ is $1-T/\epsilon$, whose magnitude at eigenvalue $1$ is $|1-1/\epsilon|$. Increasingly delicate spectral separation can amplify the remaining spectrum without bound. If two representations have the same eigenvalues for every chosen Hecke operator, no such filter separates them at all. Positivity of a packet sum then gives existence of some nonzero summand, not positivity of the prescribed one.

Why an unnormalized Euler product does not determine the scalar.

Nonzero local factors can have zero limiting product: exactly $\prod_{j=2}^m(1-1/j)=1/m$. In (30006977:scalar) the unramified factors were normalized to one; the remaining $c$ is genuinely undetermined, not made nonzero by a convergent-product argument. Its zero set can be identified with that of $L(\pi,R,1/2)$ only through global input. A functional equation with sign $+1$ also does not force a nonzero central value: $(s-1/2)^2$ is an even function with a central zero. These examples test the logic, not construct automorphic counterexamples.

Stress test.

The tensor calculation explicitly assumes one-dimensional form spaces and compatible reference vectors, and does not identify local/global measures by fiat. The filter identity assumes a discrete convergent sum; continuous or residual spectra need separate control. Unramified Hecke data may only isolate a packet. Finally, dropping the inverse $\mu$ twist changes the $L$-function itself. None of those points is discharged by saying "multiplicity one." The finite script checks the scalar and filter identities in small models, not an automorphic trace formula.

Outcome.

Outcome: Exploratory approach with unresolved gap. The attempt proves an algebraic scalar reduction and a precise spectral isolation identity, while rejecting local-test-vector nonvanishing, packet-sum positivity and nonzero-factor multiplication as substitutes for the missing theorem. A global support comparison with controlled remainder for every allowed group and fixed representation is still required. No extension of the unitary results to the full target is claimed.

Sources

• S1 Gan, Gross and Prasad, Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, §§23–24, Conjecture 24.1. https://arxiv.org/pdf/0909.2999. Catalogue formulation source.
• E1 P. Boisseau, W. Lu and H. Xue, The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups III: Proof of the main theorems, arXiv:2601.01738v1 $2026$, Theorems 1.1, 1.3, 1.4 and Sections 1.4--1.5. https://arxiv.org/html/2601.01738v1.