{"paper":{"title":"The algebra $\\mathbb{Z}_\\ell[[\\mathbb{Z}_p^d]]$ and applications to Iwasawa theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrea Bandini, Ignazio Longhi","submitted_at":"2023-12-07T19:57:58Z","abstract_excerpt":"Let $\\ell$ and $p$ be distinct primes, and let $\\G$ be an abelian pro-$p$-group. We study the structure of the algebra $\\L:=\\Z_\\ell[[\\G]]$ and of $\\L$-modules. The algebra $\\L$ turns out to be a direct product of copies of ring of integers of cyclotomic extensions of $\\Q_\\ell$ and this induces a similar decomposition for a family of $\\L$-modules. Inside this family we define Sinnott modules and provide characteristic ideals and formulas \\`a la Iwasawa for orders and ranks of their quotients. When $\\G\\simeq \\Z_p^d$\\, is the Galois group of an extension of global fields, $\\ell$-class groups and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.04666","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2312.04666/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}