{"paper":{"title":"Mean field equations, hyperelliptic curves and modular forms: II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Chin-Lung Wang","submitted_at":"2015-02-11T13:08:54Z","abstract_excerpt":"A pre-modular form $Z_n(\\sigma; \\tau)$ of weight $\\tfrac{1}{2} n(n + 1)$ is introduced for each $n \\in \\Bbb N$, where $(\\sigma, \\tau) \\in \\Bbb C \\times \\Bbb H$, such that for $E_\\tau = \\Bbb C/(\\Bbb Z + \\Bbb Z \\tau)$, every non-trivial zero of $Z_n(\\sigma; \\tau)$, namely $\\sigma \\not\\in E_\\tau[2]$, corresponds to a (scaling family of) solution to the mean field equation \\begin{equation} \\tag{MFE} \\triangle u + e^u = \\rho \\, \\delta_0 \\end{equation} on the flat torus $E_\\tau$ with singular strength $\\rho = 8\\pi n$.\n  In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve $\\bar X_n(\\tau) \\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03295","kind":"arxiv","version":3},"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"}