A Restricted-Poisson Converse Construction of Symmetric-Power Lifts for Non-CM Elliptic Curves

Version note Version v2.0. Major theorem-scope extension of v1.0r1 (DOI: 10.5281/zenodo.22867823). The core restricted-Poisson / Booker--Krishnamurthy converse induction is retained. After the induction is complete, Galois comparison and local--global compatibility upgrade the almost-all finite-place realization to an all-place symmetric-power lift for every $m \\ge 1$. The title, abstract, main theorem, notation, and analytic consequences have been revised accordingly, and Sato--Tate is now presented as a consequence of the all-place lift. The existence of symmetric-power lifts in this setting is already a special case of Newton--Thorne's theorem for regular-weight Hilbert modular forms. The contribution of v2.0 is the alternative restricted-Poisson/converse construction and its all-place recovery. Description Let $E/\\mathbb Q$ be a non-CM elliptic curve and let $\\pi \\in \\mathrm{GL}_2(\\mathbb A_{\\mathbb Q})$ be its unitary-normalized cuspidal automorphic representation. For every integer $m \\ge 1$, this preprint constructs a unitary cuspidal self-dual automorphic representation $\\Pi_m \\in \\mathrm{GL}_{m+1}(\\mathbb A_{\\mathbb Q})$, with trivial central character, whose local Langlands parameter at every place $v$ is the $m$-th symmetric power of the local parameter of $\\pi$. The construction is an alternative to the known symmetric-power existence theorem of Newton--Thorne. It proceeds by a fixed-rank induction combining a restricted three-stage Poisson transform arising from the Getz--Gu--Hsu--Leslie / Getz--Gu--Hsu geometry, the Booker--Krishnamurthy converse theorem, and the triangular Clebsch--Gordan identity $$V_2 \\otimes V_{m-2} = V_m \\oplus V_{m-2} \\oplus V_{m-4}.$$ Rankin--Selberg pole detection extracts the new block, an Archimedean local converse recovers the real parameter, and Galois irreducibility proves cuspidality. After the induction is complete, compatible Galois systems and local--global compatibility recover the ramified finite local parameters. Consequently, the completed symmetric-power $L$-functions are entire of finite order and satisfy their standard functional equations, and the non-CM Sato--Tate theorem follows. Newton--Thorne's symmetric-power theorem is not used as an input to the construction.=============================================== 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