{"paper":{"title":"The geometry of generalized Lam\\'{e} equation, I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Chang-shou Lin, Ting-Jung Kuo, Zhijie Chen","submitted_at":"2017-08-17T14:21:01Z","abstract_excerpt":"In this paper, we prove that the spectral curve $\\Gamma_{\\mathbf{n}}$ of the generalized Lam\\'{e} equation with the Treibich-Verdier potential \\begin{equation*} y^{\\prime \\prime }(z)=\\bigg[ \\sum_{k=0}^{3}n_{k}(n_{k}+1)\\wp(z+\\tfrac{% \\omega_{k}}{2}|\\tau)+B\\bigg] y(z),\\text{ \\ }n_{k}\\in \\mathbb{Z}_{\\geq0} \\end{equation*} can be embedded into the symmetric space Sym$^{N}E_{\\tau}$ of the $N$-th copy of the torus $E_{\\tau}$, where $N=\\sum n_{k}$. This embedding induces an addition map $\\sigma_{\\mathbf{n}}(\\cdot|\\tau)$ from $\\Gamma_{\\mathbf{n}}$ onto $E_{\\tau}$. The main result is to prove that the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.05306","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}