{"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"}