{"paper":{"title":"Horizontal variation of Tate--Shafarevich groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ashay A. Burungale, Haruzo Hida, Ye Tian","submitted_at":"2017-12-06T12:04:10Z","abstract_excerpt":"Let $E$ be an elliptic curve over $\\mathbb{Q}$. Let $p$ be an odd prime and $\\iota: \\overline{\\mathbb{Q}}\\hookrightarrow \\mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $\\chi$ over $K$, let $\\mathbb{Q}(\\chi)$ be the field generated by the image of $\\chi$ and $\\mathfrak{p}_{\\chi}$ the prime of $\\mathbb{Q}(\\chi)$ above $p$ determined via $\\iota_p$. Under mild hypotheses, we show that the number of class group characters $\\chi$ such that the $\\chi$-isotypic Tate--Shafarevich group of $E$ over $H_{K}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02148","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"}