{"paper":{"title":"Elliptic matroids and modular curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Matthew Baker","submitted_at":"2026-08-05T18:00:38Z","abstract_excerpt":"For $n\\geq 4$, let $T_n$ be the rank-3 matroid on $\\mathbb{Z}/n\\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible N\\'eron polygons. For $n \\geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \\mathbb{C}$, this recovers a theorem of Borisov and Roulleau"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05299","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/2608.05299/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"}