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We show that the morphism $J \\to \\mathbb{P}^{2^g-1}$ associated to the linear system $|2 \\Theta|$ may be described explicitly, for any $g \\geq 1$, using the theory of pure spinors.\n  We apply this theory to study the heights of rational points in $J(k)$, when $k$ is a number field. As a particular consequence, we show that $100\\%$ of monic, degree $2g+1$ polynomials $f(x) \\in \\mathbb{Z}[x]$ of nonzero discrimina"},"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":"2507.06865","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-07-09T14:07:04Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"59642299445472af2dc070874282ccea79a8377c327124605a28b6595ffff35c","abstract_canon_sha256":"61633b25632226a84646a191a9bc87d0143a7112164d35619afc7c7d2df22216"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:34:24.269549Z","signature_b64":"0w51BsibYdmg6ADFHzcPeiDAJiqqXM+Wg5lLV0ZjxqoPn1kRjIb8XxKcuIO/J6MBczsz7gvkVj2xSKFG984dAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4048a5121a93bc5d8af105a116c772bf186b35493065b255e2e1c004cf558e8f","last_reissued_at":"2026-07-05T11:34:24.269100Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:34:24.269100Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kummers, spinors, and heights","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Jack A. Thorne, Jef Laga","submitted_at":"2025-07-09T14:07:04Z","abstract_excerpt":"Let $f(x) = x^{2g+1} + c_1 x^{2g} + \\dots + c_{2g+1} \\in k[x]$ be a polynomial of nonzero discriminant, and let $J$ denote the Jacobian of the odd hyperelliptic curve $C : y^2 = f(x)$. We show that the morphism $J \\to \\mathbb{P}^{2^g-1}$ associated to the linear system $|2 \\Theta|$ may be described explicitly, for any $g \\geq 1$, using the theory of pure spinors.\n  We apply this theory to study the heights of rational points in $J(k)$, when $k$ is a number field. As a particular consequence, we show that $100\\%$ of monic, degree $2g+1$ polynomials $f(x) \\in \\mathbb{Z}[x]$ of nonzero discrimina"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06865","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/2507.06865/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2507.06865","created_at":"2026-07-05T11:34:24.269156+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.06865v1","created_at":"2026-07-05T11:34:24.269156+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06865","created_at":"2026-07-05T11:34:24.269156+00:00"},{"alias_kind":"pith_short_12","alias_value":"IBEKKEQ2SO6F","created_at":"2026-07-05T11:34:24.269156+00:00"},{"alias_kind":"pith_short_16","alias_value":"IBEKKEQ2SO6F3CXR","created_at":"2026-07-05T11:34:24.269156+00:00"},{"alias_kind":"pith_short_8","alias_value":"IBEKKEQ2","created_at":"2026-07-05T11:34:24.269156+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4","json":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4.json","graph_json":"https://pith.science/api/pith-number/IBEKKEQ2SO6F3CXRAWQRNR3SX4/graph.json","events_json":"https://pith.science/api/pith-number/IBEKKEQ2SO6F3CXRAWQRNR3SX4/events.json","paper":"https://pith.science/paper/IBEKKEQ2"},"agent_actions":{"view_html":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4","download_json":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4.json","view_paper":"https://pith.science/paper/IBEKKEQ2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.06865&json=true","fetch_graph":"https://pith.science/api/pith-number/IBEKKEQ2SO6F3CXRAWQRNR3SX4/graph.json","fetch_events":"https://pith.science/api/pith-number/IBEKKEQ2SO6F3CXRAWQRNR3SX4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4/action/storage_attestation","attest_author":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4/action/author_attestation","sign_citation":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4/action/citation_signature","submit_replication":"https://pith.science/pith/IBEKKEQ2SO6F3CXRAWQRNR3SX4/action/replication_record"}},"created_at":"2026-07-05T11:34:24.269156+00:00","updated_at":"2026-07-05T11:34:24.269156+00:00"}