{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:XIZRBYLXVOHVAUISP2VS3GTBU3","short_pith_number":"pith:XIZRBYLX","canonical_record":{"source":{"id":"1509.01192","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-03T18:11:32Z","cross_cats_sorted":[],"title_canon_sha256":"95852feb0ecc43666f442c5b011227ee2c3409f82c3c806c73a2a2359b45abea","abstract_canon_sha256":"b0e40d2ad312da4b8e2953eddc069e4a4a5d4030fb593d8f2a8c8e48029b8d38"},"schema_version":"1.0"},"canonical_sha256":"ba3310e177ab8f5051127eab2d9a61a6cb7a4c9aa4e82cf7f628cb2505d261d1","source":{"kind":"arxiv","id":"1509.01192","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01192","created_at":"2026-05-18T01:07:54Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01192v2","created_at":"2026-05-18T01:07:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01192","created_at":"2026-05-18T01:07:54Z"},{"alias_kind":"pith_short_12","alias_value":"XIZRBYLXVOHV","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XIZRBYLXVOHVAUIS","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XIZRBYLX","created_at":"2026-05-18T12:29:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:XIZRBYLXVOHVAUISP2VS3GTBU3","target":"record","payload":{"canonical_record":{"source":{"id":"1509.01192","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-03T18:11:32Z","cross_cats_sorted":[],"title_canon_sha256":"95852feb0ecc43666f442c5b011227ee2c3409f82c3c806c73a2a2359b45abea","abstract_canon_sha256":"b0e40d2ad312da4b8e2953eddc069e4a4a5d4030fb593d8f2a8c8e48029b8d38"},"schema_version":"1.0"},"canonical_sha256":"ba3310e177ab8f5051127eab2d9a61a6cb7a4c9aa4e82cf7f628cb2505d261d1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:54.072497Z","signature_b64":"H0FfcGkWYzt0vfsHNx6owZlQzKV6sHuYkTt5FbhrWzMsgbOmzh03eMKT2g3INdekr81sqLIeHW5S7oAkRclaBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba3310e177ab8f5051127eab2d9a61a6cb7a4c9aa4e82cf7f628cb2505d261d1","last_reissued_at":"2026-05-18T01:07:54.072118Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:54.072118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.01192","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:07:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l75PWdtoR465WfWUDlQxPiLSj4zaHBIv5Y39dFCoMY33NNnRH9CpbGdPXb40Xv8ZD2Rq4fUd6eYEHgyCku4nDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T07:23:38.820615Z"},"content_sha256":"20022b47ebbc0f0ed5dfc38132ada4a8b424bba935141457084eb8f347dae5de","schema_version":"1.0","event_id":"sha256:20022b47ebbc0f0ed5dfc38132ada4a8b424bba935141457084eb8f347dae5de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:XIZRBYLXVOHVAUISP2VS3GTBU3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal $F$-crystals and isomorphism numbers of isosimple $F$-crystals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiao Xiao","submitted_at":"2015-09-03T18:11:32Z","abstract_excerpt":"In this paper we generalize minimal $p$-divisible groups defined by Oort to $F$-crystal over an algebraically closed field of positive characteristic. We prove a structural theorem and give an explicit formula of the Frobenius endomorphism of the isosimple minimal $F$-crystals that are the building blocks of minimal $F$-crystals. We then define an invariant called the minimal height for $F$-crystals using minimal $F$-crystals and give an upper bound of the isomorphism numbers of isosimple $F$-crystals in terms of their ranks, Hodge slopes and Newton slopes."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01192","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:07:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yrqTI8GqzZkupgdxJJIshG/i7GFsQ3UbEiiimb65iqfAb6LGDktAu3ApJ9WYlLxCX/unKIlngXrCimtVOW+kDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T07:23:38.821169Z"},"content_sha256":"7b73d485bf04e6cf81e479b17722e369d93c39fe8351137a62ffe51dd9d52ee8","schema_version":"1.0","event_id":"sha256:7b73d485bf04e6cf81e479b17722e369d93c39fe8351137a62ffe51dd9d52ee8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XIZRBYLXVOHVAUISP2VS3GTBU3/bundle.json","state_url":"https://pith.science/pith/XIZRBYLXVOHVAUISP2VS3GTBU3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XIZRBYLXVOHVAUISP2VS3GTBU3/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":"2026-07-07T07:23:38Z","links":{"resolver":"https://pith.science/pith/XIZRBYLXVOHVAUISP2VS3GTBU3","bundle":"https://pith.science/pith/XIZRBYLXVOHVAUISP2VS3GTBU3/bundle.json","state":"https://pith.science/pith/XIZRBYLXVOHVAUISP2VS3GTBU3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XIZRBYLXVOHVAUISP2VS3GTBU3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:XIZRBYLXVOHVAUISP2VS3GTBU3","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":"b0e40d2ad312da4b8e2953eddc069e4a4a5d4030fb593d8f2a8c8e48029b8d38","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-03T18:11:32Z","title_canon_sha256":"95852feb0ecc43666f442c5b011227ee2c3409f82c3c806c73a2a2359b45abea"},"schema_version":"1.0","source":{"id":"1509.01192","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01192","created_at":"2026-05-18T01:07:54Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01192v2","created_at":"2026-05-18T01:07:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01192","created_at":"2026-05-18T01:07:54Z"},{"alias_kind":"pith_short_12","alias_value":"XIZRBYLXVOHV","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XIZRBYLXVOHVAUIS","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XIZRBYLX","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:7b73d485bf04e6cf81e479b17722e369d93c39fe8351137a62ffe51dd9d52ee8","target":"graph","created_at":"2026-05-18T01:07:54Z","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"},"paper":{"abstract_excerpt":"In this paper we generalize minimal $p$-divisible groups defined by Oort to $F$-crystal over an algebraically closed field of positive characteristic. We prove a structural theorem and give an explicit formula of the Frobenius endomorphism of the isosimple minimal $F$-crystals that are the building blocks of minimal $F$-crystals. We then define an invariant called the minimal height for $F$-crystals using minimal $F$-crystals and give an upper bound of the isomorphism numbers of isosimple $F$-crystals in terms of their ranks, Hodge slopes and Newton slopes.","authors_text":"Xiao Xiao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-03T18:11:32Z","title":"Minimal $F$-crystals and isomorphism numbers of isosimple $F$-crystals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01192","kind":"arxiv","version":2},"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":"sha256:20022b47ebbc0f0ed5dfc38132ada4a8b424bba935141457084eb8f347dae5de","target":"record","created_at":"2026-05-18T01:07:54Z","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":"b0e40d2ad312da4b8e2953eddc069e4a4a5d4030fb593d8f2a8c8e48029b8d38","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-03T18:11:32Z","title_canon_sha256":"95852feb0ecc43666f442c5b011227ee2c3409f82c3c806c73a2a2359b45abea"},"schema_version":"1.0","source":{"id":"1509.01192","kind":"arxiv","version":2}},"canonical_sha256":"ba3310e177ab8f5051127eab2d9a61a6cb7a4c9aa4e82cf7f628cb2505d261d1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ba3310e177ab8f5051127eab2d9a61a6cb7a4c9aa4e82cf7f628cb2505d261d1","first_computed_at":"2026-05-18T01:07:54.072118Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:54.072118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H0FfcGkWYzt0vfsHNx6owZlQzKV6sHuYkTt5FbhrWzMsgbOmzh03eMKT2g3INdekr81sqLIeHW5S7oAkRclaBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:54.072497Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.01192","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20022b47ebbc0f0ed5dfc38132ada4a8b424bba935141457084eb8f347dae5de","sha256:7b73d485bf04e6cf81e479b17722e369d93c39fe8351137a62ffe51dd9d52ee8"],"state_sha256":"89695e379ba52b53ec7bfa02a3dd039105247a7d8a67aa2c5570777fd7ed536d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cwvz5IJM7pNjVFVu4nGuhNtvF0CLw7XTOGqsidEL9dnZpgr3Y/+2Q79iGWHM2fac2Ix8bmpG9Dkzt9nkSqW9Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T07:23:38.824363Z","bundle_sha256":"c9a3f59c67fb2f907707e627a584822b65c81bbc619dee83f5d1d327c5fe0b79"}}