{"paper":{"title":"The Structure of the Group of Rational Points of an Abelian Variety over a Finite Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Caleb Springer","submitted_at":"2020-05-31T23:19:00Z","abstract_excerpt":"Let $A$ be a simple abelian variety of dimension $g$ defined over a finite field $\\mathbb{F}_q$ with Frobenius endomorphism $\\pi$. This paper describes the structure of the group of rational points $A(\\mathbb{F}_{q^n})$, for all $n \\geq 1$, as a module over the ring $R$ of endomorphisms which are defined over $\\mathbb{F}_q$, under certain technical conditions. If $[\\mathbb{Q}(\\pi) : \\mathbb{Q}]=2g$ and $R$ is a Gorenstein ring, then ${A(\\mathbb{F}_{q^n}) \\cong R/R(\\pi^n-1)}$. This includes the case when $A$ is ordinary and has maximal real multiplication. Otherwise, if $Z$ is the center of $R$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.00637","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.00637/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"}