{"paper":{"title":"Torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of the rational numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Harris B. Daniels","submitted_at":"2017-10-14T19:53:44Z","abstract_excerpt":"Let $E/\\mathbb{Q}$ be an elliptic curve and let $\\mathbb{Q}(D_4^\\infty)$ be the compositum of all extensions of $\\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we first show that $\\mathbb{Q}(D_4^\\infty)$ is in fact the compositum of all $D_4$ extensions of $\\mathbb{Q}$ and then we prove that the torsion subgroup of $E(\\mathbb{Q}(D_4^\\infty))$ is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05228","kind":"arxiv","version":4},"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"}