{"paper":{"title":"Semi-direct Galois covers of the affine line","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Bo-Hae Im, Ekin Ozman, Katherine Stevenson, Laura Hall-Seelig, Linda Gruendken, Rachel Pries","submitted_at":"2009-10-25T18:51:27Z","abstract_excerpt":"Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\\ell Z$ semi-direct product $Z/pZ$ where $\\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $\\psi:Z \\to P^1_k$ ramified only over $\\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ that admits such a cover and show that it depends only on $\\ell$, $p$, and the order $a$ of $\\ell$ modulo $p$. We also prove that the number of curves $Z$ of this minimal genus which admit such a cover is at most $(p-1)/a$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.4695","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"}