{"paper":{"title":"A theory of generalized Lam\\'e curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Generalized Lamé curves parametrize quasi-periodic solutions to elliptic equations with multiple poles and prove the Treibich conjecture for up to four symmetric pairs.","cross_cats":["math.CA"],"primary_cat":"math.AG","authors_text":"Chin-Lung Wang, Po-Sheng Wu, You-Cheng Chou","submitted_at":"2026-04-23T17:23:20Z","abstract_excerpt":"We study the generalized Lam'e equation (GLE) on an elliptic curve $E$ with multiple regular singularities $\\mathbf{p} = (p_i)_{i = 1}^r$ of weights $\\mathbf{n} = (n_i)_{i = 1}^r$. By analyzing the locus admitting quasi-periodic solutions, we construct two fundamental algebraic curves:\n  (i) The generalized Lam'e curve (GLC), $\\mathcal{Y}_{\\mathbf{n}, \\mathbf{p}}$, which lies in an affine bundle over $\\operatorname{Sym}^n E$ for total weight $n:=\\sum n_i \\in \\mathbb{Z}_{\\geq 0}$ and parametrizes generalized Hermite--Halphen ansatz solutions.\n  (ii) The log-free curve, $V_{\\mathbf{n}, \\mathbf{p"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove the Treibich conjecture stated for r=2 extra symmetric pairs, as well as its generalizations for r ≤ 4. We construct the generalized Lamé curve Y_n(p;τ) which lies in an affine bundle over Sym^n E and parametrizes generalized Hermite-Halphen ansatz solutions, and we prove that the log-free curve V_n(p;τ) is a reduced curve.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The restriction to the locus admitting solutions with quasi-periodic properties is sufficient to construct the generalized Lamé curve and to allow continuous deformation to the classical Lamé equation; if this locus is empty or the ansatz misses essential solutions for general pole configurations, the parametrization and deformation claims fail.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalized Lamé curves are built to parametrize quasi-periodic solutions of Lamé equations with multiple singularities on elliptic curves, together with a proof of the Treibich conjecture for up to four extra symmetric pairs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generalized Lamé curves parametrize quasi-periodic solutions to elliptic equations with multiple poles and prove the Treibich conjecture for up to four symmetric pairs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"709ca166040b5eb4e5d423262269c02743d0cf561872ad8ff18efaa609e2c2a7"},"source":{"id":"2604.21880","kind":"arxiv","version":2},"verdict":{"id":"83d5b245-e035-4393-a539-62898802b5a0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T20:17:29.813787Z","strongest_claim":"We prove the Treibich conjecture stated for r=2 extra symmetric pairs, as well as its generalizations for r ≤ 4. We construct the generalized Lamé curve Y_n(p;τ) which lies in an affine bundle over Sym^n E and parametrizes generalized Hermite-Halphen ansatz solutions, and we prove that the log-free curve V_n(p;τ) is a reduced curve.","one_line_summary":"Generalized Lamé curves are built to parametrize quasi-periodic solutions of Lamé equations with multiple singularities on elliptic curves, together with a proof of the Treibich conjecture for up to four extra symmetric pairs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The restriction to the locus admitting solutions with quasi-periodic properties is sufficient to construct the generalized Lamé curve and to allow continuous deformation to the classical Lamé equation; if this locus is empty or the ansatz misses essential solutions for general pole configurations, the parametrization and deformation claims fail.","pith_extraction_headline":"Generalized Lamé curves parametrize quasi-periodic solutions to elliptic equations with multiple poles and prove the Treibich conjecture for up to four symmetric pairs."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.21880/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T11:39:19.285230Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T00:34:51.445725Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"08fe28f192123ac4b77d631655bebd3c144a59814eb56085c349562c3bc6112c"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}