{"paper":{"title":"Curves of given $p$-rank with trivial automorphism group","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Darren Glass, Jeff Achter, Rachel Pries","submitted_at":"2007-08-16T13:44:54Z","abstract_excerpt":"Let $k$ be an algebraically closed field of characteristic $p >0$. Suppose $g \\geq 3$ and $0 \\leq f \\leq g$. We prove there is a smooth projective $k$-curve of genus $g$ and $p$-rank $f$ with no non-trivial automorphisms. In addition, we prove there is a smooth projective hyperelliptic $k$-curve of genus $g$ and $p$-rank $f$ whose only non-trivial automorphism is the hyperelliptic involution. The proof involves computations about the dimension of the moduli space of (hyperelliptic) $k$-curves of genus $g$ and $p$-rank $f$ with extra automorphisms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.2199","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"}