Weak Automorphy of the Triple Symmetric-Cube Tensor on $\mathrm{GL}(64)$

Let $E_1, E_2, E_3 / \mathbb Q$ be fixed non-CM elliptic curves, and let $$A_i = \operatorname{Sym}^3 \pi_i$$ be the associated automorphic representations of $\mathrm{GL}_4(\mathbb A_{\mathbb Q})$. We construct an isobaric automorphic representation $$\Pi_{333} / \mathrm{GL}_{64}(\mathbb A_{\mathbb Q})$$ whose local Langlands parameter agrees, at every finite place outside a fixed finite set, with $$\operatorname{Sym}^3 \phi_{\pi_1, v} \otimes \operatorname{Sym}^3 \phi_{\pi_2, v} \otimes \operatorname{Sym}^3 \phi_{\pi_3, v}.$$ The result is a weak automorphic transfer: no compatibility is asserted at the excluded finite places or at the Archimedean place for the final $\mathrm{GL}_{64}$ output. The proof starts from the previously established weak $\mathrm{GL}_{16}$ transfer for $A_1 \otimes A_2$. Two new ingredients are introduced. First, the missing Archimedean parameter of the weak pair transfer is recovered from highly ramified global $\mathrm{GL}_1$-twisted functional equations, using Deligne–Henniart stability together with the Archimedean local converse theorem. This implies temperedness at infinity of every cuspidal constituent of the pair output. Second, the remaining Archimedean normalization for the converse-theorem twists is calibrated at twist rank two by quadratic monomial automorphic references. A separation-prime argument removes possible constituentwise Rankin–Selberg poles, while an exact fixed-rank residual-scalar identity propagates the rank-two normalization to every twist rank $2 \le d \le 63$. The resulting completed cuspidal twists are entire of finite order and satisfy the required standard functional equations. Isobaric reassembly then supplies the full $\mathrm{GL}_{63}$ twisting family, and the converse theorem of Booker–Krishnamurthy yields the weak $\mathrm{GL}_{64}$ automorphic representation. The theorem concerns the specific triple symmetric-cube tensor parameter above; it is not a general $\mathrm{GL}_{16} \times \mathrm{GL}_4 \to \mathrm{GL}_{64}$ tensor-product functoriality theorem. Keywords automorphic representations; weak automorphy; Langlands functoriality; symmetric cube; triple tensor products; GL(64); converse theorems; Booker–Krishnamurthy converse theorem; Archimedean local converse theorem; highly ramified twists; Deligne–Henniart stability; automorphic induction; quadratic base change; Rankin–Selberg L-functions; local Langlands correspondence Recommended PDF filename Lee_WeakAutomorphy_TripleSymmetricCube_GL64_v1.0_2026.pdf Recommended source filename Lee_WeakAutomorphy_TripleSymmetricCube_GL64_v1.0_2026.tex