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Our main purpose is to investigate the structure of the torsion subgroup of $V(X)$ and its relation to the arithmetic of the curve. Using Bloch's exact sequence together with a Hasse principle for Galois cohomology arising from mod-$l$ Galois representations, we show that the mod-$l$ quotient $V(X)/lV(X)$ is governed by the mod-$l$ Galois representation on the $l$-torsion subgroup $J[l]$ of the Jacobian "},"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":"2507.22319","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-07-30T01:26:23Z","cross_cats_sorted":[],"title_canon_sha256":"8f6b2e8da696fa43f1026d08a83cb35cf9b153acbb0dc4f89a812c60826f5631","abstract_canon_sha256":"191e1a1ebb5f459a6c969c4f2bc5ebe1325332d5b0f10c962ca564bc4b4f01fa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T01:08:30.637884Z","signature_b64":"IiFgzThpTht2+YfoU3l7qAuF1cr2fjeJXOqAFdWuNdqIVN8DrRv0U2ewT8SMA2L7vsNRhWjJNWbCWJPRzrWSDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6165f67ad7c439e4227c47e54175654b5a01f66156c095ba871493da6aabac44","last_reissued_at":"2026-06-04T01:08:30.636783Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T01:08:30.636783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Hasse principle for the higher Chow groups of curves over a global field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Toshiro Hiranouchi","submitted_at":"2025-07-30T01:26:23Z","abstract_excerpt":"We study the higher Chow group $CH^2(X,1)$ of a smooth projective curve $X$ over a global field $F$, focusing on the kernel $V(X)$ of the push-forward map $CH^2(X,1) \\to CH^1(F,1) = F^\\times$. 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Using Bloch's exact sequence together with a Hasse principle for Galois cohomology arising from mod-$l$ Galois representations, we show that the mod-$l$ quotient $V(X)/lV(X)$ is governed by the mod-$l$ Galois representation on the $l$-torsion subgroup $J[l]$ of the Jacobian "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.22319","kind":"arxiv","version":3},"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/2507.22319/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":"2507.22319","created_at":"2026-06-04T01:08:30.636954+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.22319v3","created_at":"2026-06-04T01:08:30.636954+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.22319","created_at":"2026-06-04T01:08:30.636954+00:00"},{"alias_kind":"pith_short_12","alias_value":"MFS7M6WXYQ46","created_at":"2026-06-04T01:08:30.636954+00:00"},{"alias_kind":"pith_short_16","alias_value":"MFS7M6WXYQ46IIT4","created_at":"2026-06-04T01:08:30.636954+00:00"},{"alias_kind":"pith_short_8","alias_value":"MFS7M6WX","created_at":"2026-06-04T01:08:30.636954+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/MFS7M6WXYQ46IIT4I7SUC5LFJN","json":"https://pith.science/pith/MFS7M6WXYQ46IIT4I7SUC5LFJN.json","graph_json":"https://pith.science/api/pith-number/MFS7M6WXYQ46IIT4I7SUC5LFJN/graph.json","events_json":"https://pith.science/api/pith-number/MFS7M6WXYQ46IIT4I7SUC5LFJN/events.json","paper":"https://pith.science/paper/MFS7M6WX"},"agent_actions":{"view_html":"https://pith.science/pith/MFS7M6WXYQ46IIT4I7SUC5LFJN","download_json":"https://pith.science/pith/MFS7M6WXYQ46IIT4I7SUC5LFJN.json","view_paper":"https://pith.science/paper/MFS7M6WX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.22319&json=true","fetch_graph":"https://pith.science/api/pith-number/MFS7M6WXYQ46IIT4I7SUC5LFJN/graph.json","fetch_events":"https://pith.science/api/pith-number/MFS7M6WXYQ46IIT4I7SUC5LFJN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MFS7M6WXYQ46IIT4I7SUC5LFJN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MFS7M6WXYQ46IIT4I7SUC5LFJN/action/storage_attestation","attest_author":"https://pith.science/pith/MFS7M6WXYQ46IIT4I7SUC5LFJN/action/author_attestation","sign_citation":"https://pith.science/pith/MFS7M6WXYQ46IIT4I7SUC5LFJN/action/citation_signature","submit_replication":"https://pith.science/pith/MFS7M6WXYQ46IIT4I7SUC5LFJN/action/replication_record"}},"created_at":"2026-06-04T01:08:30.636954+00:00","updated_at":"2026-06-04T01:08:30.636954+00:00"}