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We show that there is an isomorphism\n  \\det(\\pi_*\\CL)^{\\o times 24}\\simeq\\big(\\pi_*\\omega_{\\CA}^{\\vee}\\big)^{\\o times 12d}\n  of line bundles on $S$, where $d$ is the rank of the (locally free) sheaf $\\pi_*\\CL$. We also show that the numbers 24 and $12d$ are sharp in the following sense: if $N>1$ is a common divisor of 12 and 24, then there are data as above such that \n  \\det(\\pi_*\\CL)^{\\o times (24/"},"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":"math/0611105","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"2006-11-04T18:06:04Z","cross_cats_sorted":[],"title_canon_sha256":"886a0f225dd0b15944ece503c4eae5df61383c9dea2dcd58c24c1bdd0eeea902","abstract_canon_sha256":"8305742a34157819eea2c79ad9979f8052fa2f91d7db2f283f4b2b56aa1d6924"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:31.114464Z","signature_b64":"iZ21mgIZURNOgtH6Fj9dpFH4UtDNmnLcHy4uxMa2OtCk/hk65i5rCOeDnEBTQUnNFUSP64GTHYo3TGPGiVY9BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11179fda5a1e5911d63cc67a6320ffaf912f664ba2b88e7b99be13da5126e492","last_reissued_at":"2026-05-18T03:02:31.113350Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:31.113350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the determinant bundles of abelian schemes","license":"","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Damian R\\\"ossler, Vincent Maillot","submitted_at":"2006-11-04T18:06:04Z","abstract_excerpt":"Let $\\pi:\\CA\\ra S$ be an abelian scheme over a scheme $S$ which is quasi-projective over an affine noetherian scheme and let $\\CL$ be a symmetric, rigidified, relatively ample line bundle on $\\CA$. We show that there is an isomorphism\n  \\det(\\pi_*\\CL)^{\\o times 24}\\simeq\\big(\\pi_*\\omega_{\\CA}^{\\vee}\\big)^{\\o times 12d}\n  of line bundles on $S$, where $d$ is the rank of the (locally free) sheaf $\\pi_*\\CL$. We also show that the numbers 24 and $12d$ are sharp in the following sense: if $N>1$ is a common divisor of 12 and 24, then there are data as above such that \n  \\det(\\pi_*\\CL)^{\\o times (24/"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611105","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"math/0611105","created_at":"2026-05-18T03:02:31.113437+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0611105v2","created_at":"2026-05-18T03:02:31.113437+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611105","created_at":"2026-05-18T03:02:31.113437+00:00"},{"alias_kind":"pith_short_12","alias_value":"CELZ7WS2DZMR","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"CELZ7WS2DZMRDVR4","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"CELZ7WS2","created_at":"2026-05-18T12:25:53.939244+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/CELZ7WS2DZMRDVR4YZ5GGIH7V6","json":"https://pith.science/pith/CELZ7WS2DZMRDVR4YZ5GGIH7V6.json","graph_json":"https://pith.science/api/pith-number/CELZ7WS2DZMRDVR4YZ5GGIH7V6/graph.json","events_json":"https://pith.science/api/pith-number/CELZ7WS2DZMRDVR4YZ5GGIH7V6/events.json","paper":"https://pith.science/paper/CELZ7WS2"},"agent_actions":{"view_html":"https://pith.science/pith/CELZ7WS2DZMRDVR4YZ5GGIH7V6","download_json":"https://pith.science/pith/CELZ7WS2DZMRDVR4YZ5GGIH7V6.json","view_paper":"https://pith.science/paper/CELZ7WS2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0611105&json=true","fetch_graph":"https://pith.science/api/pith-number/CELZ7WS2DZMRDVR4YZ5GGIH7V6/graph.json","fetch_events":"https://pith.science/api/pith-number/CELZ7WS2DZMRDVR4YZ5GGIH7V6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CELZ7WS2DZMRDVR4YZ5GGIH7V6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CELZ7WS2DZMRDVR4YZ5GGIH7V6/action/storage_attestation","attest_author":"https://pith.science/pith/CELZ7WS2DZMRDVR4YZ5GGIH7V6/action/author_attestation","sign_citation":"https://pith.science/pith/CELZ7WS2DZMRDVR4YZ5GGIH7V6/action/citation_signature","submit_replication":"https://pith.science/pith/CELZ7WS2DZMRDVR4YZ5GGIH7V6/action/replication_record"}},"created_at":"2026-05-18T03:02:31.113437+00:00","updated_at":"2026-05-18T03:02:31.113437+00:00"}