{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:4BWLOEFZTA7ICSMQHN4GUU7LYX","short_pith_number":"pith:4BWLOEFZ","schema_version":"1.0","canonical_sha256":"e06cb710b9983e8149903b786a53ebc5ea6f532d0f3164df41c22af3689f76b3","source":{"kind":"arxiv","id":"1005.2998","version":1},"attestation_state":"computed","paper":{"title":"Remarks on the Fourier coefficients of modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Kirti Joshi","submitted_at":"2010-05-17T19:11:10Z","abstract_excerpt":"We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\\Q$ which is not $\\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that $N_p(E)=#E(\\F_p)=p+1-a_p(E)$ is also a prime. We consider a variant of this question. For a newform $f$, without CM, of weight $k\\geq 4$, on $\\Gamma_0(M)$ with trivial Nebentypus $\\chi_0$ and with integer Fourier coefficients, let $N_p(f)=\\chi_0(p)p^{k-1}+1-a_p(f)$ (here $a_p(f)$ is the $p^{th}$-Fourier coefficient of $f$). We show under GRH and Artin's Holomorphy Conjecture "},"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":"1005.2998","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-05-17T19:11:10Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"0976b6564e524f6adb7092011da5b8d6292f60cf44cf787053ba1a9b76cc90e4","abstract_canon_sha256":"cccc8fbe6cd61b8c74a16e4b6d2e2b35e83c4721decc7be892a4d58d5534b72d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:08.832660Z","signature_b64":"PaR25T2z0ywEHHsMrTDZP6YA/E2H/x4sAtbseedyZCvuOoN8rpDU0eVqivZ/aNvP3lnB40+znBxGroZwme0fBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e06cb710b9983e8149903b786a53ebc5ea6f532d0f3164df41c22af3689f76b3","last_reissued_at":"2026-05-18T03:21:08.831859Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:08.831859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Remarks on the Fourier coefficients of modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Kirti Joshi","submitted_at":"2010-05-17T19:11:10Z","abstract_excerpt":"We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\\Q$ which is not $\\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that $N_p(E)=#E(\\F_p)=p+1-a_p(E)$ is also a prime. We consider a variant of this question. For a newform $f$, without CM, of weight $k\\geq 4$, on $\\Gamma_0(M)$ with trivial Nebentypus $\\chi_0$ and with integer Fourier coefficients, let $N_p(f)=\\chi_0(p)p^{k-1}+1-a_p(f)$ (here $a_p(f)$ is the $p^{th}$-Fourier coefficient of $f$). We show under GRH and Artin's Holomorphy Conjecture "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.2998","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":""},"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":"1005.2998","created_at":"2026-05-18T03:21:08.831971+00:00"},{"alias_kind":"arxiv_version","alias_value":"1005.2998v1","created_at":"2026-05-18T03:21:08.831971+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.2998","created_at":"2026-05-18T03:21:08.831971+00:00"},{"alias_kind":"pith_short_12","alias_value":"4BWLOEFZTA7I","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_16","alias_value":"4BWLOEFZTA7ICSMQ","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_8","alias_value":"4BWLOEFZ","created_at":"2026-05-18T12:26:03.138858+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/4BWLOEFZTA7ICSMQHN4GUU7LYX","json":"https://pith.science/pith/4BWLOEFZTA7ICSMQHN4GUU7LYX.json","graph_json":"https://pith.science/api/pith-number/4BWLOEFZTA7ICSMQHN4GUU7LYX/graph.json","events_json":"https://pith.science/api/pith-number/4BWLOEFZTA7ICSMQHN4GUU7LYX/events.json","paper":"https://pith.science/paper/4BWLOEFZ"},"agent_actions":{"view_html":"https://pith.science/pith/4BWLOEFZTA7ICSMQHN4GUU7LYX","download_json":"https://pith.science/pith/4BWLOEFZTA7ICSMQHN4GUU7LYX.json","view_paper":"https://pith.science/paper/4BWLOEFZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1005.2998&json=true","fetch_graph":"https://pith.science/api/pith-number/4BWLOEFZTA7ICSMQHN4GUU7LYX/graph.json","fetch_events":"https://pith.science/api/pith-number/4BWLOEFZTA7ICSMQHN4GUU7LYX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4BWLOEFZTA7ICSMQHN4GUU7LYX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4BWLOEFZTA7ICSMQHN4GUU7LYX/action/storage_attestation","attest_author":"https://pith.science/pith/4BWLOEFZTA7ICSMQHN4GUU7LYX/action/author_attestation","sign_citation":"https://pith.science/pith/4BWLOEFZTA7ICSMQHN4GUU7LYX/action/citation_signature","submit_replication":"https://pith.science/pith/4BWLOEFZTA7ICSMQHN4GUU7LYX/action/replication_record"}},"created_at":"2026-05-18T03:21:08.831971+00:00","updated_at":"2026-05-18T03:21:08.831971+00:00"}