{"paper":{"title":"The geometry of generalized Lam\\'{e} equation, II: Existence of pre-modular forms and application","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":"2018-07-20T08:57:25Z","abstract_excerpt":"In this paper, the second in a series, we continue to study 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),\\quad n_{k}\\in \\mathbb{Z}_{\\geq0} \\end{equation*} from the monodromy aspect. We prove the existence of a pre-modular form $Z_{r,s}^{\\mathbf{n}}(\\tau)$ of weight $\\frac{1}{2}\\sum n_k(n_k+1)$ such that the monodromy data $(r,s)$ is characterized by $Z_{r,s}^{\\mathbf{n}}(\\tau)=0$. This generalizes the result in \\cite{LW2}, where the Lam\\'{e} case ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.07745","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"}