A Restricted-Poisson Converse Route to the Non-CM Sato–Tate Theorem

Version v1.0r1. Minor source-explicitness revision of v1.0. This revision clarifies the bibliographic identification of GGHL Version 3, makes the correspondence between Lemma A.8 and the defining Archimedean $S^{\\mathrm{as}}$-seminorms of GGH more explicit, and refines one auxiliary zero/pole-locus statement in Section 10. No theorem statement, proof architecture, normalization convention, or main conclusion has changed. ============================= Description This preprint develops an alternative automorphic route to the non-CM Sato–Tate theorem for elliptic curves over $\\mathbb Q$. Let $E/\\mathbb Q$ be a non-CM elliptic curve and let $\\pi$ be its unitary-normalized automorphic representation of $\\mathrm{GL}_2(\\mathbb A_{\\mathbb Q})$. For every $m \\ge 1$, we construct a unitary cuspidal automorphic representation $\\Pi_m$ of $\\mathrm{GL}_{m+1}(\\mathbb A_{\\mathbb Q})$, with trivial central character and self-duality, whose unramified finite local parameters agree with the $m$-th symmetric powers of those of $\\pi$ at all but finitely many finite places and whose real parameter agrees exactly with the symmetric-power parameter. The construction proceeds by a fixed-rank induction using a restricted three-stage Poisson transform, the Booker–Krishnamurthy converse theorem, and the Clebsch–Gordan identity $$\\operatorname{Sym}^2 \\otimes \\operatorname{Sym}^{m-2} = \\operatorname{Sym}^m \\oplus \\operatorname{Sym}^{m-2} \\oplus \\operatorname{Sym}^{m-4}.$$ Rankin–Selberg pole detection extracts the new symmetric-power block; highly ramified stability and an Archimedean local converse recover the real parameter; and a Galois-theoretic irreducibility argument establishes cuspidality. The resulting induction gives holomorphy and nonvanishing at $s=1$ for the incomplete symmetric-power $L$-functions and yields the non-CM Sato–Tate theorem through Serre’s equidistribution criterion. The argument does not assume the conjectural ambient GGHL Poisson formula and does not use all-symmetric-power automorphy as an input. Keywords Sato–Tate theorem; symmetric-power L-functions; automorphic representations; converse theorem; Poisson summation; Rankin–Selberg L-functions; non-CM elliptic curves Related identifier Identifier: 10.5281/zenodo.22852677 Identifier scheme: DOI Relation: References Resource type: Publication / Preprint Related work: Lee Byoungwoo, “A Conditional Weak $\\mathrm{GL}(12)$ Transfer for the $(2,3)$ Symmetric-Power Tensor,” Version 3.1. Additional description — Note / Version note Version v1.0. First public preprint release. This record contains the 63-page source-sealed manuscript prepared for journal submission on 21 September 2026. The manuscript incorporates the final auxiliary-clearing, local-normalization, semantic-typing, and source-dependency revisions completed prior to public release. No conjectural ambient GGHL Poisson formula or all-symmetric-power automorphy theorem is assumed. 2020 Mathematics Subject Classification Primary: 11F70 Secondary: 11F66, 11G05