{"paper":{"title":"The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christopher Skinner, Dimitar Jetchev, Xin Wan","submitted_at":"2015-12-21T22:26:35Z","abstract_excerpt":"Let $E/\\mathbb{Q}$ be a semistable elliptic curve such that $\\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\\mathbb{Q}$ for each prime $p \\geq 5$ of good reduction such that $E[p]$ is irreducible: $$ \\mathrm{ord}_p \\left (\\frac{L'(E,1)}{\\Omega_E\\cdot\\mathrm{Reg}(E/\\mathbb{Q})} \\right ) = \\mathrm{ord}_p \\left (\\#\\mathrm{Sha}(E/\\mathbb{Q})\\prod_{\\ell\\leq \\infty} c_\\ell(E/\\mathbb{Q}) \\right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06894","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"}