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

Abstract

Version 2.1 is the version submitted to the Journal of the London Mathematical Society on 22 September 2026. Relative to Version 2.0, this release incorporates the final journal-facing revisions, including further clarification of the fixed-rank restricted-Poisson construction, local and Archimedean normalization, the converse-theorem interface, and the post-construction local-global compatibility argument. The positioning relative to the Newton-Thorne symmetric-power automorphy theorem is also stated explicitly. The main theorem and overall mathematical scope are unchanged: the paper gives an alternative restricted-Poisson/converse construction of all symmetric-power lifts for non-CM elliptic curves over Q, with Newton-Thorne used for contextual comparison rather than as an input to the proof. This version is the frozen JLMS submission version. 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