{"paper":{"title":"On Frobenius semisimplicity in Hida families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jyoti Prakash Saha","submitted_at":"2018-01-09T18:25:57Z","abstract_excerpt":"Let $p\\geq 5$ be a prime and $\\ell\\neq p$ be a prime not dividing the tame level of a $p$-ordinary Hida family. We prove that the actions of the Frobenius element at $\\ell$ on the Galois representations attached to almost all arithmetic specializations are semisimple and non-scalar. If $k_f$ denotes the weight of a cusp form $f(z)= \\sum_{n\\geq 1} a_\\ell(f) e^{2\\pi i n z}$, then the inequality $$|a_\\ell(f) | \\leq 2 \\ell^{(k_f-1)/2},$$ predicted by the Ramanujan conjecture, is a strict inequality for almost all members $f$ of the family."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03070","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"}