{"paper":{"title":"On the distribution of critical points of the Eisenstein series $E_6$ and monodromy interpretation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhijie Chen","submitted_at":"2025-06-04T01:10:36Z","abstract_excerpt":"In previous works joint with Lin, we proved that the Eisenstein series $E_4$ (resp. $E_2$) has at most one critical point in every fundamental domain $\\gamma(F_0)$ of $\\Gamma_{0}(2)$, where $\\gamma(F_0)$ are translates of the basic fundamental domain $F_0$ via the M\\\"{o}bius transformation of $\\gamma\\in\\Gamma_{0}(2)$. But the method can not work for the Eisenstein series $E_6$.\n  In this paper, we develop a new approach to show that $E_6'(\\tau)$ has exactly either $1$ or $2$ zeros in every fundamental domain $\\gamma(F_0)$ of $\\Gamma_{0}(2)$. A criterion for $\\gamma(F_0)$ containing exactly $2$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.03475","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.03475/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}