Boundary Completion and Coherent Finite Tensor Closure for Weak Automorphic Transfers
Year: 2026
Type: preprint
Abstract: This paper completes the remaining rank-two boundary in a fixed-rank weak tensor-transfer construction for general linear groups and derives a coherent finite tensor-closure principle. The new argument treats the boundary case \(q=2\) by decomposing the isobaric first factor according to the ranks of its cuspidal constituents. Rank-one constituents are handled by ordinary Rankin–Selberg theory. Higher-rank constituents are treated by reorienting the previously established hybrid functional equation so that the higher-rank constituent supplies the genuine axis, together with a finite-family LCM balancing argument for the companion central characters. Rank-two constituents are handled separately through the classical automorphic tensor product on \(\mathrm{GL}_2 \times \mathrm{GL}_2\), followed by highly ramified finite-place comparison and stable-twist Archimedean recovery. Combined with the previously established iteration for \(q\ge 3\) and the elementary character-twist case \(q=1\), this yields a boundary-free weak tensor iteration for every fixed positive rank. Iteration gives unitary isobaric weak automorphic realizations of arbitrary fixed finite tensor products under a common unramified-tempered-place hypothesis. Strong multiplicity one then shows that the resulting global isomorphism class is independent of auxiliary choices, permutations of the factors, and parenthesization, giving a coherent finite tensor-closure operation at the level of global isomorphism classes. As an arithmetic application, every fixed finite tensor product of symmetric powers attached to non-CM elliptic curves admits such a weak automorphic realization. In particular, the previously excluded all-boundary triple \((1,1,1)\) is recovered on \(\mathrm{GL}_8\). All results are fixed-rank. The transfer is weak at a finite exceptional set, with exact compatibility at the Archimedean place. No uniformity in the number or ranks of the tensor factors is claimed, and the proof does not assume a conjectural ambient total Poisson formula. Version v1.0.
Author: Lee, Byoungwoo
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214456505
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