A Unconditional Weak $\mathrm{GL}(12)$ Transfer for the $(2,3)$ Symmetric-Power Tensor (v3.1)

Abstract

We construct a weak automorphic realization on $\mathrm{GL}_{12}(\mathbb A_{\mathbb Q})$ of the tensor parameter $$\operatorname{Sym}^2(\pi_i) \otimes \operatorname{Sym}^3(\pi_j),$$ where $\pi_i$ and $\pi_j$ arise from two non-CM elliptic curves over $\mathbb Q$. More precisely, we construct an isobaric automorphic representation $\Pi_{23}$ of $\mathrm{GL}_{12}(\mathbb A_{\mathbb Q})$ such that $$\phi_{\Pi_{23,v}} = \phi_{\operatorname{Sym}^2(\pi_i),v} \otimes \phi_{\operatorname{Sym}^3(\pi_j),v}$$ at all but finitely many finite places $v$. The proof is organized around the Booker–Krishnamurthy converse theorem. For twists of ranks $3 \le d \le 11$, a representation-scoped restricted Poisson construction for mixed triples $(3,4,d)$ supplies the required good finite Euler factors, continuation, finite-part entirety, finite order, and functional equations without invoking the full ambient GGHL Poisson conjecture. At the bad finite places, the normalization is obtained from a split Piatetski–Shapiro–Rallis one-data calibration. After exact Chen–Sun contraction and removal of the known pair gamma factors, GGH moving-frame covariance reduces the compact sharp-frame family to a reference frame. The remaining functional is identified with the raw Siegel–Radon/PSR intertwiner, and the degree-one and degree-two Getz–Hsu–Leslie Tate convolutions recover the canonical PSR normalization. The degree-two augmentation is adelically neutral, while exact cross-rank normalization shows that no additional ambient-rank scalar appears for $3 \le d \le 11$. The Archimedean normalization is calibrated unconditionally by a diagonal comparison. Writing $A = \pi_i$, the Clebsch–Gordan identity and the trivial determinant of the elliptic-curve representation give $$\operatorname{Sym}^2 A \otimes \operatorname{Sym}^3 A \simeq \operatorname{Sym}^5 A \boxplus \operatorname{Sym}^3 A \boxplus A$$ at the level of local Langlands parameters. By symmetric-power automorphy, the right-hand side provides an automorphic $\mathrm{GL}_{12}$ calibration object with the expected local parameters at every place. Since weight-two elliptic-curve representations have the same real Langlands parameter, the diagonal and original tensor problems have identical Archimedean local data in rank three. Comparison of the restricted functional equation with the standard automorphic Rankin–Selberg functional equation therefore determines the standard rank-three real gamma factor, which is then propagated to all required ranks by the exact cross-rank normalization. Together with the rank-one and rank-two twists, the converse theorem yields the stated weak $\mathrm{GL}_{12}$ transfer. The weak automorphy statement is unconditional. Under the separate hypothesis that the incomplete self Rankin–Selberg $L$-function attached to the good-place parameter has a simple pole at $s = 1$, the resulting automorphic representation is cuspidal. Version 3.0 replaces the packet-level triality hypothesis used in Version 2.0 by the unconditional diagonal Archimedean calibration described above. It also incorporates the completed finite-place PSR moving-frame calibration, exact cross-rank normalization, finite-conductor comparison, and final submission-facing audits of normalization, notation, proof dependencies, and theorem scope. No claim is made of full ramified local compatibility or of general $\mathrm{GL}_3 \times \mathrm{GL}_4 \to \mathrm{GL}_{12}$ functoriality. ================================= Version v3.1 — verification-hardened submission revision. The main theorem and its scope are unchanged from v3.0: the paper proves an unconditional weak automorphic realization on GL(12) for the (2,3)(2,3) symmetric-power tensor at almost all finite places; the cuspidality conclusion remains conditional on the separate simple self Rankin–Selberg pole hypothesis. This revision substantially strengthens verification and exposition. It adds independent appendices for the Booker–Krishnamurthy converse-theorem hypotheses, the diagonal Archimedean calibration, and cross-rank scalar rigidity; hardens the mixed-rank covariance and restricted Poisson source interfaces; clarifies the raw/corrected/residual/canonical normalization layers; improves notation and cold-reader accessibility; and expands the historical and methodological bibliography. No stronger ramified local compatibility or general GL(3) × GL(4) functoriality statement is claimed.