{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:3DNT2LK6PN4562P5XN3XRYFDYO","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":"6c878150f99a102975ba6047669c70d71ae38b6f5801c44f1bf8a1fd74507873","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-22T17:52:12Z","title_canon_sha256":"767ba957ea4e4b30964c50f87a77f536456532d17c7c8f517428867d9a9e17dc"},"schema_version":"1.0","source":{"id":"1308.4934","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.4934","created_at":"2026-05-18T03:15:19Z"},{"alias_kind":"arxiv_version","alias_value":"1308.4934v1","created_at":"2026-05-18T03:15:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.4934","created_at":"2026-05-18T03:15:19Z"},{"alias_kind":"pith_short_12","alias_value":"3DNT2LK6PN45","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3DNT2LK6PN4562P5","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3DNT2LK6","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:1caeeeab21c3256ff3d45bb9182ff86c3af42501caa4211fc5f1ef8c8858d660","target":"graph","created_at":"2026-05-18T03:15:19Z","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":"Let $G=\\Sp(2g,\\mathbb{Z})$ be the symplectic group over the integers. Given $m\\in \\mathbb{N}$, it is natural to ask if there exists a non-trivial matrix $A\\in G$ such that $A^{m}=I$, where $I$ is the identity matrix in $G$. In this paper, we determine the possible values of $m\\in \\mathbb{N}$ for which the above problem has a solution. We also show that there is an upper bound on the maximal order of an element in $G$. As an illustration, we apply our results to the group $\\Sp(4,\\mathbb{Z})$ and determine the possible orders of elements in it. Finally, we use a presentation of $\\Sp(4,\\mathbb{Z}","authors_text":"Ganesh Ji Omar, Kumar Balasubramanian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-22T17:52:12Z","title":"Some remarks on the symplectic group $Sp(2g, \\mathbb{Z})$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4934","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":"sha256:26da838bea4e45485ee2b9fa4447a9e775ecc112dc8d4a6831da4761a9684721","target":"record","created_at":"2026-05-18T03:15:19Z","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":"6c878150f99a102975ba6047669c70d71ae38b6f5801c44f1bf8a1fd74507873","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-22T17:52:12Z","title_canon_sha256":"767ba957ea4e4b30964c50f87a77f536456532d17c7c8f517428867d9a9e17dc"},"schema_version":"1.0","source":{"id":"1308.4934","kind":"arxiv","version":1}},"canonical_sha256":"d8db3d2d5e7b79df69fdbb7778e0a3c39acb14d586f30ceaebfdc5cb7ffa0a51","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8db3d2d5e7b79df69fdbb7778e0a3c39acb14d586f30ceaebfdc5cb7ffa0a51","first_computed_at":"2026-05-18T03:15:19.270339Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:19.270339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qo1bvrDXZ94KP0p7I0xeDjB8rOTCQDlNtjpDqNp6fLBk6/vMEJTtG3haSf5u4Ayv3csXmMpgfBqCSPZGRNhwAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:19.271227Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.4934","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:26da838bea4e45485ee2b9fa4447a9e775ecc112dc8d4a6831da4761a9684721","sha256:1caeeeab21c3256ff3d45bb9182ff86c3af42501caa4211fc5f1ef8c8858d660"],"state_sha256":"a0b0409be1b5f1a2d5e98e6e01641883e6e532517f99ed68914c2e9d8c4b9836"}