A Conditional Weak $\mathrm{GL}(12)$ Transfer for the $(2,3)$ Symmetric-Power Tensor (v2.0)

Abstract

We construct a conditional 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 non-CM elliptic curves over $\mathbb Q$. The argument is designed around the Booker–Krishnamurthy converse theorem. For twists of ranks $3 \le d \le 11$, we establish a representation-scoped restricted Poisson construction for mixed triples $(3,4,d)$, yielding the required good finite Euler factors, meromorphic continuation, finite-part entirety, finite order, and functional equations without invoking the full ambient GGHL Poisson conjecture. The finite bad-place normalization is obtained by 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 resulting functional is identified with the raw Siegel–Radon/PSR intertwiner, while the degree-one and degree-two Getz–Hsu–Leslie Tate convolutions recover the canonical PSR normalization. The degree-two augmentation is adelically neutral. Exact cross-rank normalization shows that no new ambient-rank scalar appears for $3 \le d \le 11$. At the real place, the rank-three gamma factor is calibrated by a virtual quotient of standard automorphic Rankin–Selberg $L$-functions using a selected packet-level $\mathrm{GL}_8$ triality input, and the normalization is then propagated to all required ranks. Together with the rank-one and rank-two twists, the converse theorem produces an isobaric automorphic representation on $\mathrm{GL}_{12}$ matching the expected tensor parameter at all but finitely many finite places. Under a separate simple self Rankin–Selberg pole hypothesis, the resulting representation is cuspidal. Version 2.0 incorporates the completed finite-place PSR moving-frame calibration, the exact cross-rank normalization, the finite-conductor rank-three comparison needed for the Archimedean calibration, and a final referee-facing audit of normalization layers, notation, proof dependencies, and scope. No claim of full ramified local compatibility, full $\mathrm{GL}_3 \times \mathrm{GL}_4$ functoriality, or the ambient GGHL total Fourier theorem is made.