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The set of $v$'s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field.\n  We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $\\Delta(v)$ of $End(E(v)"},"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":"1509.07095","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-23T18:57:49Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"9ccff7cafd94f3627d2e74eeebe9b5b4341209e93f77319fde4ade73c63ac1a3","abstract_canon_sha256":"8ba2a3353f1d2ff65b2aea8feed7eef97fd440a3b84936006e9a738a7a5c2c3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:35.274485Z","signature_b64":"Njdb0KQzPU4zcohNRPm+IDCuAM/qwG0xU23NGP6Kr3bai09gxxVnUe5sHDvyFWPLiLQSfB05rR1PEJhc7EPnCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28b1e00bf60b81bff1aa261aa99b3a18fe0c252d4c7055774d42c2376d4471a2","last_reissued_at":"2026-05-18T01:19:35.273986Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:35.273986Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Endomorphism rings of reductions of elliptic curves and abelian varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Yuri G. Zarhin","submitted_at":"2015-09-23T18:57:49Z","abstract_excerpt":"Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$'s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field.\n  We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $\\Delta(v)$ of $End(E(v)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07095","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":"1509.07095","created_at":"2026-05-18T01:19:35.274051+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.07095v2","created_at":"2026-05-18T01:19:35.274051+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07095","created_at":"2026-05-18T01:19:35.274051+00:00"},{"alias_kind":"pith_short_12","alias_value":"FCY6AC7WBOA3","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"FCY6AC7WBOA374NK","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"FCY6AC7W","created_at":"2026-05-18T12:29:19.899920+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/FCY6AC7WBOA374NKEYNKTGZ2DD","json":"https://pith.science/pith/FCY6AC7WBOA374NKEYNKTGZ2DD.json","graph_json":"https://pith.science/api/pith-number/FCY6AC7WBOA374NKEYNKTGZ2DD/graph.json","events_json":"https://pith.science/api/pith-number/FCY6AC7WBOA374NKEYNKTGZ2DD/events.json","paper":"https://pith.science/paper/FCY6AC7W"},"agent_actions":{"view_html":"https://pith.science/pith/FCY6AC7WBOA374NKEYNKTGZ2DD","download_json":"https://pith.science/pith/FCY6AC7WBOA374NKEYNKTGZ2DD.json","view_paper":"https://pith.science/paper/FCY6AC7W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.07095&json=true","fetch_graph":"https://pith.science/api/pith-number/FCY6AC7WBOA374NKEYNKTGZ2DD/graph.json","fetch_events":"https://pith.science/api/pith-number/FCY6AC7WBOA374NKEYNKTGZ2DD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FCY6AC7WBOA374NKEYNKTGZ2DD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FCY6AC7WBOA374NKEYNKTGZ2DD/action/storage_attestation","attest_author":"https://pith.science/pith/FCY6AC7WBOA374NKEYNKTGZ2DD/action/author_attestation","sign_citation":"https://pith.science/pith/FCY6AC7WBOA374NKEYNKTGZ2DD/action/citation_signature","submit_replication":"https://pith.science/pith/FCY6AC7WBOA374NKEYNKTGZ2DD/action/replication_record"}},"created_at":"2026-05-18T01:19:35.274051+00:00","updated_at":"2026-05-18T01:19:35.274051+00:00"}