{"paper":{"title":"Residual Representations of Semistable Principally Polarized Abelian Varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pedro Lemos, Samir Siksek, Samuele Anni","submitted_at":"2015-08-02T09:54:59Z","abstract_excerpt":"Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\\ell$ be a prime and let $\\bar{\\rho}_{A,\\ell} : G_{\\mathbb{Q}} \\rightarrow \\mathrm{GSp}_{2d}(\\mathbb{F}_\\ell)$ be the representation giving the action of $G_{\\mathrm{Q}} :=\\mathrm{Gal}(\\bar{\\mathrm{Q}}/\\mathrm{Q})$ on the $\\ell$-torsion group $A[\\ell]$. We show that if $\\ell \\ge \\max(5,d+2)$, and if image of $\\bar{\\rho}_{A,\\ell}$ contains a transvection then $\\bar{\\rho}_{A,\\ell}$ is either reducible or surjective.\n  With the help of this we study surjectivity of $\\bar{\\rho}_{A,\\ell}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00211","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"}