{"paper":{"title":"The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alina Carmen Cojocaru, Andrew Michael Shulman","submitted_at":"2013-04-08T04:42:07Z","abstract_excerpt":"Let $\\psi$ be a generic Drinfeld module of rank $r \\geq 2$. We study the first elementary divisor $d_{1, \\wp}(\\psi)$ of the reduction of $\\psi$ modulo a prime $\\wp$, as $\\wp$ varies. In particular, we prove the existence of the density of the primes $\\wp$ for which $d_{1, \\wp} (\\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\\psi$ modulo $\\wp$ and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.2100","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":""},"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"}