{"paper":{"title":"Genus growth in $\\mathbb{Z}_p$-towers of function fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Daqing Wan, Michiel Kosters","submitted_at":"2017-03-15T23:12:25Z","abstract_excerpt":"Let $K$ be a function field over a finite field $k$ of characteristic $p$ and let $K_{\\infty}/K$ be a geometric extension with Galois group $\\mathbb{Z}_p$. Let $K_n$ be the corresponding subextension with Galois group $\\mathbb{Z}/p^n\\mathbb{Z}$ and genus $g_n$. In this paper, we give a simple explicit formula $g_n$ in terms of an explicit Witt vector construction of the $\\mathbb{Z}_p$-tower. This formula leads to a tight lower bound on $g_n$ which is quadratic in $p^n$. Furthermore, we determine all $\\mathbb{Z}_p$-towers for which the genus sequence is stable, in the sense that there are $a,b,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05420","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":""},"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"}