{"paper":{"title":"Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\\ell}+y^{2m}=z^p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Samir Siksek, Samuele Anni","submitted_at":"2015-06-09T10:56:49Z","abstract_excerpt":"Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \\nmid n$ and $n \\ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular.\n  Let $\\ell$, $m$, $p$ be prime, with $\\ell$, $m \\ge 5$ and $p \\ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\\mathbb{Q}(\\zeta_p)^+$, and s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02860","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"}