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To each non-degenerate rational plane $L$ in the four-dimensional quadratic space $(\\mathbb{Q}^4,Q)$ we can naturally attach a periodic geodesic on the Bianchi orbifold $\\mathrm{SL}_2(\\mathbb{Z}[i])\\backslash \\mathbb{H}^3$ which records the position of $L$ in the Grassmannian up to integer rotations. Moreover, each such plane $L$ defines a CM point and a periodic geodesic on the modular curve through restriction of $Q$ to $L$ and its orthogonal complement. Lastly, the local isomorphism between $\\mathrm{SO}_{1,3}"},"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":"2606.03472","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-02T10:49:33Z","cross_cats_sorted":[],"title_canon_sha256":"8244a3e9e5cc20591986d606ed45d2c611661bf47b4483c0979ef5b29e838a67","abstract_canon_sha256":"b41ad4291250602b46f27dcd3186e3c75b58c88b6d8b5163a07afa148dbe8a15"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:58.773748Z","signature_b64":"GfHGb9BghfCULnmUHZhSaokVZkQSo5pCzvd95LFHGtHbdtAtuolwn9vjeasBOAdPaER7pumRfSx+30FpQ1dXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11115af98a731364dafa6337e40acc2a96dfba586136f683a1470c7c3a835cb8","last_reissued_at":"2026-06-03T01:05:58.773370Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:58.773370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Planes in quadratic 4-space and associated shapes of lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Wieser, Konstantin Andritsch, Menny Aka","submitted_at":"2026-06-02T10:49:33Z","abstract_excerpt":"Let $Q=-x_1^1-x_2^2-x_3^2+x_4^2$ be the standard signature $(1,3)$ quadratic form. To each non-degenerate rational plane $L$ in the four-dimensional quadratic space $(\\mathbb{Q}^4,Q)$ we can naturally attach a periodic geodesic on the Bianchi orbifold $\\mathrm{SL}_2(\\mathbb{Z}[i])\\backslash \\mathbb{H}^3$ which records the position of $L$ in the Grassmannian up to integer rotations. Moreover, each such plane $L$ defines a CM point and a periodic geodesic on the modular curve through restriction of $Q$ to $L$ and its orthogonal complement. 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