{"paper":{"title":"Quadratic Twists of Elliptic Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"John Coates, Shuai Zhai, Ye Tian, Yongxiong Li","submitted_at":"2013-12-13T17:25:29Z","abstract_excerpt":"In this paper, we show that Tian's induction method can be generalised to study the Birch-Swinnerton-Dyer conjecture for the quadratic twists, both with global root number $+1$ and with global root number $-1$, of certain elliptic curves $E$ defined over $\\mathbb Q$. In particular, for the curve $E = X_0(49)$ we prove the following results. Let $q_1, \\ldots, q_r$ be distinct primes which are congruent to $1$ modulo $4$ and inert in the field $F = \\mathbb Q(\\sqrt{-7})$, and let $E^{(R)}$ be the twist of $E$ by the quadratic extension $\\mathbb Q(\\sqrt{R})/\\mathbb Q$, where $R=q_1\\ldots q_r$. The"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3884","kind":"arxiv","version":3},"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"}