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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 $\\bar{\\rho}_{A,\\ell}$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.00211","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-02T09:54:59Z","cross_cats_sorted":[],"title_canon_sha256":"1b6e1838edb477635688384a87f70801e94988ea9f88f61a2bfb75181701bad5","abstract_canon_sha256":"4488ee9d9423ee9c936a640badffc7b451e1bdf12cc9153178d69050c1f83bf0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:25.515206Z","signature_b64":"aVbbE6Aw8txJbkR6cmcJfQS14Hvtes/3m8NaZlM+uUvO8mwsk7naXz26nsTI57iI0OF1fHlbv4hthznXhIQvCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a64220b1db5ff14051419e31caf3be16bd615320f259611de4ef11bb77fb0019","last_reissued_at":"2026-05-18T01:17:25.514495Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:25.514495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1508.00211","created_at":"2026-05-18T01:17:25.514614+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.00211v2","created_at":"2026-05-18T01:17:25.514614+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.00211","created_at":"2026-05-18T01:17:25.514614+00:00"},{"alias_kind":"pith_short_12","alias_value":"UZBCBMO3L7YU","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UZBCBMO3L7YUAUKB","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UZBCBMO3","created_at":"2026-05-18T12:29:44.643036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2","json":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2.json","graph_json":"https://pith.science/api/pith-number/UZBCBMO3L7YUAUKBTYY4V456C2/graph.json","events_json":"https://pith.science/api/pith-number/UZBCBMO3L7YUAUKBTYY4V456C2/events.json","paper":"https://pith.science/paper/UZBCBMO3"},"agent_actions":{"view_html":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2","download_json":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2.json","view_paper":"https://pith.science/paper/UZBCBMO3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.00211&json=true","fetch_graph":"https://pith.science/api/pith-number/UZBCBMO3L7YUAUKBTYY4V456C2/graph.json","fetch_events":"https://pith.science/api/pith-number/UZBCBMO3L7YUAUKBTYY4V456C2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2/action/storage_attestation","attest_author":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2/action/author_attestation","sign_citation":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2/action/citation_signature","submit_replication":"https://pith.science/pith/UZBCBMO3L7YUAUKBTYY4V456C2/action/replication_record"}},"created_at":"2026-05-18T01:17:25.514614+00:00","updated_at":"2026-05-18T01:17:25.514614+00:00"}