Hybrid Restricted Triple-Product Functional Equations and Weak Tensor Transfers for Fixed Symmetric-Power Pairs

Abstract

Version 1.2 is a notation- and exposition-hardened revision of Version 1.1. The main theorems, mathematical scope, and proof architecture are unchanged. The principal revisions are as follows: the spherical Satake parameter and the spherical Satake isomorphism are now notationally separated, removing a semantic type collision in the good-place Hecke-shell construction; the local normalized-induction datum in the cross-rank scalar-rigidity argument is fully typed, including the representations, companion rank, contraction variables, Levi coordinates, and transpose-inverse notation; several local notation collisions and use-before-definition issues are removed, including the target-variable notation in the Booker–Krishnamurthy reassembly, the double-boundary local zeta notation, the auxiliary-tube/BPLZZ notation, and the Gaussian radial parameter; source-dependent notation used in the hybrid stabilizer, pure-center character, finite local sectors, and Archimedean calibration is clarified; project-internal shorthand in the normalization discussion is reduced or explicitly explained; the double-boundary specialization is rewritten with explicit rank-pattern notation and reader-facing terminology. These revisions are intended to make the manuscript more self-contained and referee-facing. They do not alter the hybrid restricted triple-product functional equation, the universal split-PSR contraction, the finite or Archimedean normalization arguments, or the resulting weak tensor-transfer theorem. The manuscript remains 35 pages. =========================================================== Version 1.1 strengthens two proof interfaces without changing the main theorem, scope, or proof architecture. The revision makes the finite-stage cutoff transfer in the good-place coefficient-one argument explicit and self-contained, thereby clarifying how the stabilized Hecke-shell germ is realized by a compact restricted section. It also expands the Archimedean calibration argument to spell out the passage from limit multiplicity and the residual-spectrum classification to density on the tempered slice, followed by continuity and holomorphic uniqueness on the corresponding maximal totally real locus. A short real-parameter identification in the diagonal calibration is added, and the bibliography is synchronized with the current public version of the companion single-curve symmetric-power manuscript. No theorem statement or principal conclusion has changed. ========================================================== This preprint develops a representation-scoped hybrid form of the restricted triple-product method for arbitrary fixed ranks $r_1, r_2, r_3 \ge 2$, with at least one rank at least three. Higher-rank axes are treated by the affine-$\Psi$ transforms of Getz--Gu--Hsu, while each rank-two axis is replaced by the ordinary normalized $\mathrm{GL}_2$ Whittaker--Mellin functional equation. The construction retains the rank-independent outer oscillator and projective Lagrangian geometry of Getz--Gu--Hsu--Leslie, establishes an exact coefficient-one unramified formula, gives one- and two-boundary global reassembly, and contracts all hybrid patterns to a common split Piatetski--Shapiro--Rallis core. The analytic package includes auxiliary-only continuation clearing, canonical finite-place normalization, a Clebsch--Gordan real calibration, one fixed highly ramified character for the converse-theorem twist family, and constituentwise reassembly for the Booker--Krishnamurthy converse theorem. The argument does not invoke the conjectural total GGHL Poisson formula. As an application, assume weak cuspidal single-curve symmetric-power realizations $A / \mathrm{GL}_{a+1}(\mathbb A_{\mathbb Q})$ and $B / \mathrm{GL}_{b+1}(\mathbb A_{\mathbb Q})$. For $N=(a+1)(b+1)$, the paper constructs an isobaric automorphic representation $\Pi_{a,b} / \mathrm{GL}_N(\mathbb A_{\mathbb Q})$ whose local parameter equals $$\operatorname{Sym}^a\phi_{\pi_i,v} \otimes \operatorname{Sym}^b\phi_{\pi_j,v}$$ at every finite place outside a fixed finite set. Combined with the companion single-curve symmetric-power realizations, this yields weak tensor transfers for every fixed symmetric-power pair attached to two non-CM elliptic curves. The conclusion is deliberately weak: no compatibility at ramified finite places or at the real place is asserted for the final converse-theorem output.