{"paper":{"title":"Every finite abelian group is the group of rational points of an ordinary abelian variety over $\\mathbb{F}_2$, $\\mathbb{F}_3$ and $\\mathbb{F}_5$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Caleb Springer, Stefano Marseglia","submitted_at":"2021-05-17T19:30:10Z","abstract_excerpt":"We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\\mathbb{F}_2$, $\\mathbb{F}_3$ and $\\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite field $\\mathbb{F}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over $\\mathbb{F}_q$ when $q$ is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over $\\mathbb{F}_2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.08125","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/2105.08125/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"}