Depth-Three Sato-Tate Harmonics for Three Non-CM Elliptic Curves: Analytic Closure of the Final Triple-Cubic Mode

Abstract

This paper studies the final triple-cubic mode in the finite depth-three Sato–Tate harmonic packet attached to three non-CM elliptic curves over $\mathbb{Q}$. Let $\pi_i$ denote the unitary cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ associated with $E_i$. The target is the degree-$64$ Euler product $$L_{333}(u)=L\!\left(u,\operatorname{Sym}^3\pi_1\otimes\operatorname{Sym}^3\pi_2\otimes\operatorname{Sym}^3\pi_3\right).$$ Under the standing pairwise independence hypothesis (H1), we prove that $L_{333}(u)$ admits meromorphic continuation to $\mathbb{C}$ and is holomorphic and nonzero on $$\operatorname{Re}(u)\ge 1.$$ The analytic argument develops a restricted rank-$(4,4,4)$ form of the fiber-bundle Poisson strategy of Getz–Gu–Hsu–Leslie, combined with the Fourier–Poisson theory for affine $\Psi$-bundles of Getz–Gu–Hsu. On an explicit global restricted test class, we establish the source-to-dual three-stage transform, the required Fourier covariance, absolute projective-height reassembly, coefficient-one good-unramified local data, prescribed-point nonvanishing at exceptional places, and the Archimedean asymptotic-Schwartz and two-sided rapidity conditions required for Mellin continuation. These ingredients yield analytic continuation and edge pole-freeness for $L_{333}$ without assuming the ambient GGHL Poisson conjecture. Edge nonvanishing is obtained independently from an exact nonnegative character polynomial. Its character expansion has $$c_{333}=1,\qquadc_{000}=\frac{3949}{4000},\qquad1-c_{000}=\frac{51}{4000}>0,$$ while every other nonconstant term is supported on at most two elliptic factors and is controlled by established symmetric-power and Rankin–Selberg theory. Positivity of logarithmic-derivative coefficients excludes a zero at $u=1$, and a de la Vallée Poussin amplifier, with $$4-3c_{000}=\frac{4153}{4000}>0,$$ gives zero-freeness on the full edge $\operatorname{Re}(u)=1$. Assuming in addition the prior edge hypothesis (H2) for the other $62$ nontrivial depth-three modes, the result completes the finite $64$-member depth-three harmonic packet. The result is a finite-packet analytic theorem. It does **not** assert unrestricted Sato–Tate equidistribution, the ambient GGHL Poisson conjecture, or an automorphic $\mathrm{GL}_{64}$ triple-product transfer.