{"paper":{"title":"On Greenberg's $L$-invariant of the symmetric sixth power of an ordinary cusp form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Robert Harron","submitted_at":"2010-07-13T21:56:12Z","abstract_excerpt":"We derive a formula for Greenberg's $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\\geq4$, under some technical assumptions. This requires a \"sufficiently rich\" Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to $\\GSp(4)_{/\\QQ}$ of Ramakrishnan--Shahidi and the Hida theory of this group developed by Tilouine--Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg's $L$-invariant of all"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2213","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"}