{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:4EJOGLPLSEVR3C53GTWBXXUTZE","short_pith_number":"pith:4EJOGLPL","canonical_record":{"source":{"id":"2608.01595","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-08-03T01:59:49Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"2b583074925d25400e60a0f15d4b2e6a68a30a328c75e93b614c95703dbe730d","abstract_canon_sha256":"3a9a17d3bc8de1fa54aceb3c3140bd56ccc5228988bc1ff2bfea1532022e376a"},"schema_version":"1.0"},"canonical_sha256":"e112e32deb912b1d8bbb34ec1bde93c91258ca443970a35ef359dd17afcd96b4","source":{"kind":"arxiv","id":"2608.01595","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.01595","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"arxiv_version","alias_value":"2608.01595v1","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01595","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"pith_short_12","alias_value":"4EJOGLPLSEVR","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"pith_short_16","alias_value":"4EJOGLPLSEVR3C53","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"pith_short_8","alias_value":"4EJOGLPL","created_at":"2026-08-04T02:06:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:4EJOGLPLSEVR3C53GTWBXXUTZE","target":"record","payload":{"canonical_record":{"source":{"id":"2608.01595","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-08-03T01:59:49Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"2b583074925d25400e60a0f15d4b2e6a68a30a328c75e93b614c95703dbe730d","abstract_canon_sha256":"3a9a17d3bc8de1fa54aceb3c3140bd56ccc5228988bc1ff2bfea1532022e376a"},"schema_version":"1.0"},"canonical_sha256":"e112e32deb912b1d8bbb34ec1bde93c91258ca443970a35ef359dd17afcd96b4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:06:05.871160Z","signature_b64":"r9kQRK6dx60VAtc8op63dzR2tU+nOu2WfPMNh54WRwyZ7cg39itNLk4vli/1CE8Ct05Nn501Tf5ceGIwS+myCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e112e32deb912b1d8bbb34ec1bde93c91258ca443970a35ef359dd17afcd96b4","last_reissued_at":"2026-08-04T02:06:05.869642Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:06:05.869642Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2608.01595","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-08-04T02:06:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vwu35QSqNbES0y0N4fAbGRaBOgeNKRlI7GTCt9VbMUThDZSO+oaZC4ZI6+lDf8cfr73axdjGooy/qBQsWm+VDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T14:03:06.671004Z"},"content_sha256":"cb48e9478a5506552a4ee65595c8814d5fc91cb15b3036a37d06c7bedb47defc","schema_version":"1.0","event_id":"sha256:cb48e9478a5506552a4ee65595c8814d5fc91cb15b3036a37d06c7bedb47defc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:4EJOGLPLSEVR3C53GTWBXXUTZE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Finite abelian subgroups of algebraic groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AG","authors_text":"Danny Ofek, Federico Scavia, Zinovy Reichstein","submitted_at":"2026-08-03T01:59:49Z","abstract_excerpt":"Let $k$ be an algebraically closed field, and let $G$ be an algebraic $k$-group. We study finite abelian $k$-subgroups $A \\subset G$ whose order is not divisible by the characteristic of $k$. This is a classical topic in the theory of algebraic groups going back to the work of Borel in the early 1960s. We sharpen previously known results on the structure of $A$. In particular, we show that there exists a maximal torus $T$ of $G$ such that the index $[A: (A \\cap T)]$ divides the Grothendieck torsion index $t(G)$. We also show that there exists a maximal torus $T$ such that the quotient group $A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01595","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/2608.01595/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-08-04T02:06:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gMVYETk0R/2uv8oi89dLra4bUJWIFO7S36kZcMHQvV1kfST6Xu0Lk+gakL9Dl1kLTRgVMMA6Z78W/XIxxitKCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T14:03:06.671512Z"},"content_sha256":"8065a4d73d79f44facb235fe124770b88462d19e6d93ee4af8789bca7028439d","schema_version":"1.0","event_id":"sha256:8065a4d73d79f44facb235fe124770b88462d19e6d93ee4af8789bca7028439d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4EJOGLPLSEVR3C53GTWBXXUTZE/bundle.json","state_url":"https://pith.science/pith/4EJOGLPLSEVR3C53GTWBXXUTZE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4EJOGLPLSEVR3C53GTWBXXUTZE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-19T14:03:06Z","links":{"resolver":"https://pith.science/pith/4EJOGLPLSEVR3C53GTWBXXUTZE","bundle":"https://pith.science/pith/4EJOGLPLSEVR3C53GTWBXXUTZE/bundle.json","state":"https://pith.science/pith/4EJOGLPLSEVR3C53GTWBXXUTZE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4EJOGLPLSEVR3C53GTWBXXUTZE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4EJOGLPLSEVR3C53GTWBXXUTZE","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3a9a17d3bc8de1fa54aceb3c3140bd56ccc5228988bc1ff2bfea1532022e376a","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-08-03T01:59:49Z","title_canon_sha256":"2b583074925d25400e60a0f15d4b2e6a68a30a328c75e93b614c95703dbe730d"},"schema_version":"1.0","source":{"id":"2608.01595","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.01595","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"arxiv_version","alias_value":"2608.01595v1","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01595","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"pith_short_12","alias_value":"4EJOGLPLSEVR","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"pith_short_16","alias_value":"4EJOGLPLSEVR3C53","created_at":"2026-08-04T02:06:05Z"},{"alias_kind":"pith_short_8","alias_value":"4EJOGLPL","created_at":"2026-08-04T02:06:05Z"}],"graph_snapshots":[{"event_id":"sha256:8065a4d73d79f44facb235fe124770b88462d19e6d93ee4af8789bca7028439d","target":"graph","created_at":"2026-08-04T02:06:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.01595/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $k$ be an algebraically closed field, and let $G$ be an algebraic $k$-group. We study finite abelian $k$-subgroups $A \\subset G$ whose order is not divisible by the characteristic of $k$. This is a classical topic in the theory of algebraic groups going back to the work of Borel in the early 1960s. We sharpen previously known results on the structure of $A$. In particular, we show that there exists a maximal torus $T$ of $G$ such that the index $[A: (A \\cap T)]$ divides the Grothendieck torsion index $t(G)$. We also show that there exists a maximal torus $T$ such that the quotient group $A","authors_text":"Danny Ofek, Federico Scavia, Zinovy Reichstein","cross_cats":["math.GR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-08-03T01:59:49Z","title":"Finite abelian subgroups of algebraic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01595","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:cb48e9478a5506552a4ee65595c8814d5fc91cb15b3036a37d06c7bedb47defc","target":"record","created_at":"2026-08-04T02:06:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3a9a17d3bc8de1fa54aceb3c3140bd56ccc5228988bc1ff2bfea1532022e376a","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-08-03T01:59:49Z","title_canon_sha256":"2b583074925d25400e60a0f15d4b2e6a68a30a328c75e93b614c95703dbe730d"},"schema_version":"1.0","source":{"id":"2608.01595","kind":"arxiv","version":1}},"canonical_sha256":"e112e32deb912b1d8bbb34ec1bde93c91258ca443970a35ef359dd17afcd96b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e112e32deb912b1d8bbb34ec1bde93c91258ca443970a35ef359dd17afcd96b4","first_computed_at":"2026-08-04T02:06:05.869642Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-04T02:06:05.869642Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r9kQRK6dx60VAtc8op63dzR2tU+nOu2WfPMNh54WRwyZ7cg39itNLk4vli/1CE8Ct05Nn501Tf5ceGIwS+myCQ==","signature_status":"signed_v1","signed_at":"2026-08-04T02:06:05.871160Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.01595","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb48e9478a5506552a4ee65595c8814d5fc91cb15b3036a37d06c7bedb47defc","sha256:8065a4d73d79f44facb235fe124770b88462d19e6d93ee4af8789bca7028439d"],"state_sha256":"605a32a5d2c71516abe24e0f9e0e513c1a4222e95d922679a3ca36f9a952de5d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5MbvrbDHAc2WJI0oyaN96mTlLLNHj56LvqGIbj62ggvkXj2hxY9Y7HIJtJ+sseGy+xCIC7xf1p+nbcx/8z3iCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T14:03:06.676398Z","bundle_sha256":"389d5743353f1a002bedde1af686c741057e31c61eaecf9463191c8fe174d15d"}}