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In this setting, it is known that $L(E,s)$ possesses an analytic continuation to $\\mathbb C$. The period of $E$ can be written (up to a power of $2$) as the product of the Tamagawa numbers of $E$ with $\\Omega_E/\\sqrt{|\\Delta_K|}$, where $\\Omega_E$ is a quantity, independent of $\\omega_E$, which encodes the real periods of $E$ when $K$ is real and the covolume of the "},"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":"1511.09001","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-29T12:22:16Z","cross_cats_sorted":[],"title_canon_sha256":"be060f84fc4871540b968c6419c311dfbdd13daf91bf4a7a357b9200e1d3f325","abstract_canon_sha256":"5d220cd5532e4d757c1eee71d9351a21f86df4b5967b1549c50085eda358664a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:10.798354Z","signature_b64":"InzUAUagQ29or7i97GqzvwCUSNZVQ39lIReIEdB6TWOT6QQPq/sYHSy78n0ZOVpUW+ACbLYxPk58hs1NZrBLAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"711937a160bc213b038585f332d3a145a706a78ec66061deb094ede4039b7780","last_reissued_at":"2026-05-18T00:35:10.797938Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:10.797938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On $L$-functions of quadratic $\\mathbb{Q}$-curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrea Ferraguti, Peter Bruin","submitted_at":"2015-11-29T12:22:16Z","abstract_excerpt":"Let $K$ be a quadratic number field of discriminant $\\Delta_K$, let $E$ be a $\\mathbb Q$-curve without CM completely defined over $K$ and let $\\omega_E$ be an invariant differential on $E$. Let $L(E,s)$ be the $L$-function of $E$. In this setting, it is known that $L(E,s)$ possesses an analytic continuation to $\\mathbb C$. The period of $E$ can be written (up to a power of $2$) as the product of the Tamagawa numbers of $E$ with $\\Omega_E/\\sqrt{|\\Delta_K|}$, where $\\Omega_E$ is a quantity, independent of $\\omega_E$, which encodes the real periods of $E$ when $K$ is real and the covolume of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09001","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":"1511.09001","created_at":"2026-05-18T00:35:10.797992+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.09001v2","created_at":"2026-05-18T00:35:10.797992+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09001","created_at":"2026-05-18T00:35:10.797992+00:00"},{"alias_kind":"pith_short_12","alias_value":"OEMTPILAXQQT","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"OEMTPILAXQQTWA4F","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"OEMTPILA","created_at":"2026-05-18T12:29:34.919912+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/OEMTPILAXQQTWA4FQXZTFU5BIW","json":"https://pith.science/pith/OEMTPILAXQQTWA4FQXZTFU5BIW.json","graph_json":"https://pith.science/api/pith-number/OEMTPILAXQQTWA4FQXZTFU5BIW/graph.json","events_json":"https://pith.science/api/pith-number/OEMTPILAXQQTWA4FQXZTFU5BIW/events.json","paper":"https://pith.science/paper/OEMTPILA"},"agent_actions":{"view_html":"https://pith.science/pith/OEMTPILAXQQTWA4FQXZTFU5BIW","download_json":"https://pith.science/pith/OEMTPILAXQQTWA4FQXZTFU5BIW.json","view_paper":"https://pith.science/paper/OEMTPILA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.09001&json=true","fetch_graph":"https://pith.science/api/pith-number/OEMTPILAXQQTWA4FQXZTFU5BIW/graph.json","fetch_events":"https://pith.science/api/pith-number/OEMTPILAXQQTWA4FQXZTFU5BIW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OEMTPILAXQQTWA4FQXZTFU5BIW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OEMTPILAXQQTWA4FQXZTFU5BIW/action/storage_attestation","attest_author":"https://pith.science/pith/OEMTPILAXQQTWA4FQXZTFU5BIW/action/author_attestation","sign_citation":"https://pith.science/pith/OEMTPILAXQQTWA4FQXZTFU5BIW/action/citation_signature","submit_replication":"https://pith.science/pith/OEMTPILAXQQTWA4FQXZTFU5BIW/action/replication_record"}},"created_at":"2026-05-18T00:35:10.797992+00:00","updated_at":"2026-05-18T00:35:10.797992+00:00"}