{"paper":{"title":"Critical $L$-values for some quadratic twists of Gross curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrzej D\\k{a}browski, Lucjan Szymaszkiewicz, Tomasz J\\k{e}drzejak","submitted_at":"2019-04-18T11:13:10Z","abstract_excerpt":"Let $K=\\Bbb Q(\\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\\beta)}$ denote its quadratic twist, with $\\beta=\\sqrt{-q}$. The curve $E$ is defined over the Hilbert class field $H$ of $K$. We use Magma to calculate the values $L(E/H,1)$ for all such $q$'s up to some reasonable ranges (different for $q\\equiv 7 \\, \\text{mod} \\, 8$ and $q\\equiv 3 \\, \\text{mod} \\, 8$). All these values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we can calculate hypothetical orders of $\\sza(E/H)$ in these cases. Our calculations "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.08691","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"}