{"paper":{"title":"On Unitary Monodromy of Second-Order Ordinary Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.CA","authors_text":"Alex Zitzewitz, David Darrow, Eric Chen","submitted_at":"2024-12-10T21:28:05Z","abstract_excerpt":"Given a second-order, holomorphic, linear differential equation $Lf=0$ on a punctured Riemann surface, we say that its monodromy group $G\\subset\\operatorname{GL}(2,\\mathbb{C})$ is `unitary' if it preserves a non-degenerate Hermitian form $H$ on $\\mathbb{C}^2$ under the action $g\\circ H=g^\\dagger H g$. In the present work, we give two sets of necessary and sufficient conditions for a monodromy group $G\\subset\\operatorname{GL}(2,\\mathbb{C})$ to be unitary. First, in the case that the natural representation of $G$ on $\\mathbb{C}^2$ is irreducible, we show that unitarity is equivalent to a set of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.07932","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.07932/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"}