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We propose a new type of random alternating matrix model $M_{*,\\mathbf t}^{\\mathrm{Alt}}(\\mathbb{F}_2)$ over $\\mathbb{F}_2$ with 0, 1 or 2 ``holes'', with associated Markov chains, described by parameter $\\mathbf t=(t_1,\\cdots,t_s)\\in\\mathbb{Z}^s$ where $s$ is the number of ``holes''. We proved that for each equivalence classes of quadratic twists of elliptic curves:\n  (1) The distribution of 2-Selmer ranks agrees with the distribution of coranks of matrices"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2503.21462","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NT","submitted_at":"2025-03-27T12:52:42Z","cross_cats_sorted":[],"title_canon_sha256":"e173fcbeb1def0cba5821fb0364107769d787578f1407ecb5115e1b6bf005be0","abstract_canon_sha256":"3ef9281537ecdcfbef8e2416931f037fed801971717e61373c10d7d8a262a4b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:40:25.822425Z","signature_b64":"ebkbdmrNFdSHByKLLoPTP1KnYWvvgphiMo+bxuq8wZzn2NwfHsFu6zcHT7TsdO2JUoLOUC9LCI8FL8Ed4ZFbDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd5f156d29c1057ee2cc7e84c55d6d0ba0b528bbe160092258b260c10c544a2b","last_reissued_at":"2026-07-05T10:40:25.821932Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:40:25.821932Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Distribution of 2-Selmer ranks of Quadratic Twists of Elliptic Curves over $\\mathbb{Q}$","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinzhao Pan, Ye Tian","submitted_at":"2025-03-27T12:52:42Z","abstract_excerpt":"We characterize the distribution of 2-Selmer ranks of quadratic twists of elliptic curves over $\\mathbb{Q}$ with full rational 2-torsion. 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