{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:IULUZX33RTFQQLD2WV4WK2YEPS","short_pith_number":"pith:IULUZX33","schema_version":"1.0","canonical_sha256":"45174cdf7b8ccb082c7ab579656b047caf33e97a77a70f68f97a4adf6a8f7aeb","source":{"kind":"arxiv","id":"1612.02845","version":2},"attestation_state":"computed","paper":{"title":"The 1-eigenspace for matrices in $\\operatorname{GL}_2(\\mathbb{Z}_\\ell)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonella Perucca, Davide Lombardo","submitted_at":"2016-12-08T21:27:54Z","abstract_excerpt":"Fix some prime number $\\ell$ and consider an open subgroup $G$ either of $\\operatorname{GL}_2(\\mathbb{Z}_\\ell)$ or of the normalizer of a Cartan subgroup of $\\operatorname{GL}_2(\\mathbb{Z}_\\ell)$. The elements of $G$ act on $(\\mathbb{Z}/\\ell^n \\mathbb{Z})^2$ for every $n\\geqslant 1$ and hence also on the direct limit, and we call 1-eigenspace the group of fixed points. We partition $G$ by considering the possible group structures for the 1-eigenspace and show how to evaluate with a finite procedure the Haar measure of all sets in the partition. The results apply to all elliptic curves defined "},"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":"1612.02845","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-08T21:27:54Z","cross_cats_sorted":[],"title_canon_sha256":"7866b8858989c80becec46b441007db851c945449c7cb256a4c7d40ca175c40d","abstract_canon_sha256":"0f597acf21707f558b44b3c34e2c34eac323c34435b002a2437b49eef056d820"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:46.388133Z","signature_b64":"5ls6Y5H73GaK7nm5RgtN+joZ9ub094wBuRKTo9/9UfowXYvfTFLLyvUTELgScKp+EPO1+HeKP8kooGQVGK80DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45174cdf7b8ccb082c7ab579656b047caf33e97a77a70f68f97a4adf6a8f7aeb","last_reissued_at":"2026-05-18T00:38:46.387462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:46.387462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The 1-eigenspace for matrices in $\\operatorname{GL}_2(\\mathbb{Z}_\\ell)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonella Perucca, Davide Lombardo","submitted_at":"2016-12-08T21:27:54Z","abstract_excerpt":"Fix some prime number $\\ell$ and consider an open subgroup $G$ either of $\\operatorname{GL}_2(\\mathbb{Z}_\\ell)$ or of the normalizer of a Cartan subgroup of $\\operatorname{GL}_2(\\mathbb{Z}_\\ell)$. The elements of $G$ act on $(\\mathbb{Z}/\\ell^n \\mathbb{Z})^2$ for every $n\\geqslant 1$ and hence also on the direct limit, and we call 1-eigenspace the group of fixed points. We partition $G$ by considering the possible group structures for the 1-eigenspace and show how to evaluate with a finite procedure the Haar measure of all sets in the partition. The results apply to all elliptic curves defined "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02845","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":"1612.02845","created_at":"2026-05-18T00:38:46.387560+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.02845v2","created_at":"2026-05-18T00:38:46.387560+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02845","created_at":"2026-05-18T00:38:46.387560+00:00"},{"alias_kind":"pith_short_12","alias_value":"IULUZX33RTFQ","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"IULUZX33RTFQQLD2","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"IULUZX33","created_at":"2026-05-18T12:30:22.444734+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/IULUZX33RTFQQLD2WV4WK2YEPS","json":"https://pith.science/pith/IULUZX33RTFQQLD2WV4WK2YEPS.json","graph_json":"https://pith.science/api/pith-number/IULUZX33RTFQQLD2WV4WK2YEPS/graph.json","events_json":"https://pith.science/api/pith-number/IULUZX33RTFQQLD2WV4WK2YEPS/events.json","paper":"https://pith.science/paper/IULUZX33"},"agent_actions":{"view_html":"https://pith.science/pith/IULUZX33RTFQQLD2WV4WK2YEPS","download_json":"https://pith.science/pith/IULUZX33RTFQQLD2WV4WK2YEPS.json","view_paper":"https://pith.science/paper/IULUZX33","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.02845&json=true","fetch_graph":"https://pith.science/api/pith-number/IULUZX33RTFQQLD2WV4WK2YEPS/graph.json","fetch_events":"https://pith.science/api/pith-number/IULUZX33RTFQQLD2WV4WK2YEPS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IULUZX33RTFQQLD2WV4WK2YEPS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IULUZX33RTFQQLD2WV4WK2YEPS/action/storage_attestation","attest_author":"https://pith.science/pith/IULUZX33RTFQQLD2WV4WK2YEPS/action/author_attestation","sign_citation":"https://pith.science/pith/IULUZX33RTFQQLD2WV4WK2YEPS/action/citation_signature","submit_replication":"https://pith.science/pith/IULUZX33RTFQQLD2WV4WK2YEPS/action/replication_record"}},"created_at":"2026-05-18T00:38:46.387560+00:00","updated_at":"2026-05-18T00:38:46.387560+00:00"}