{"paper":{"title":"Mean field equations, hyperelliptic curves and modular forms: I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Ching-Li Chai, Chin-Lung Wang","submitted_at":"2015-02-11T13:11:20Z","abstract_excerpt":"We develop a theory connecting the following three areas: (a) the mean field equation (MFE) $\\triangle u + e^u = \\rho\\, \\delta_0$, $\\rho \\in \\mathbb R_{>0}$ on flat tori $E_\\tau = \\mathbb C/(\\mathbb Z + \\mathbb Z\\tau)$, (b) the classical Lam\\'e equations and (c) modular forms. A major theme in part I is a classification of developing maps $f$ attached to solutions $u$ of the mean field equation according to the type of transformation laws (or monodromy) with respect to $\\Lambda$ satisfied by $f$.\n  We are especially interested in the case when the parameter $\\rho$ in the mean field equation is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03297","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"}