{"paper":{"title":"Critical points of the classical Eisenstein series of weight two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chang-shou Lin, Zhijie Chen","submitted_at":"2017-07-16T01:32:17Z","abstract_excerpt":"In this paper, we completely determine the critical points of the normalized Eisenstein series $E_2(\\tau)$ of weight $2$. Although $E_2(\\tau)$ is not a modular form, our result shows that $E_2(\\tau)$ has at most one critical point in every fundamental domain of $\\Gamma_{0}(2)$. We also give a criteria for a fundamental domain containing a critical point of $E_2(\\tau)$. Furthermore, under the M\\\"obius transformation of $\\Gamma_{0}(2)$ action, all critical points can be mapped into the basic fundamental domain $F_0$ and their images are contained densely on three smooth curves. A geometric inter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04804","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"}