{"paper":{"title":"Remarks on a theorem of Silverman","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Khai-Hoan Nguyen-Dang, Quang-Khai Nguyen","submitted_at":"2026-07-10T00:11:13Z","abstract_excerpt":"Motivated by a theorem of Silverman, we consider the following problem. Let $A$ be an abelian variety over a global field $K$. Given a non-torsion point $P \\in A(K)$, for a sufficiently large positive integer $n$, whether there exists a place $v$ of $K$ such that the order of the reduction of $P$ modulo $v$ is $n$? In this article, we first show that this holds for an elliptic curve over a global function field of positive characteristic $p>3$ and for sufficiently large positive integers $n$ coprime to $p$. In the second part of the paper, we consider its relative version over $\\mathbb C$. Mor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09002","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.09002/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"}