{"paper":{"title":"The growth of the rank of Abelian varieties upon extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Filip Najman, Peter Bruin","submitted_at":"2012-10-22T23:05:07Z","abstract_excerpt":"We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields.\n  First, we show that if $L/K$ is a finite Galois extension of number fields such that $\\Gal(L/K)$ does not have an index 2 subgroup and $A/K$ is an Abelian variety, then $\\rk A(L)-\\rk A(K)$ can never be 1. We obtain more precise results when $\\Gal(L/K)$ is of odd order, alternating, $\\SL_2(\\F_p)$ or $\\PSL_2(\\F_p)$. This implies a restriction on $\\rk E(K(E[p]))-\\rk E(K(\\zeta_p))$ when $E/K$ is an elliptic curve whose mod $p$ Galois representation is surjective. Similar re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6085","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"}