{"paper":{"title":"Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dylan Laird, Sam Frengley","submitted_at":"2026-02-23T13:13:12Z","abstract_excerpt":"We exhibit examples of geometrically simple abelian surfaces $A/\\mathbb{Q}$ with conductor bounded by $(10\\,000)^2$ whose Tate--Shafarevich groups contain a subgroup isomorphic to $(\\mathbb{Z}/p\\mathbb{Z})^2$ for each $p = 5, 7, 11, 13$. To find these examples we generalise work of Cremona--Freitas to give a candidate list of all congruences of a certain type between pairs of weight $2$ newforms $f \\in S_2^{\\mathrm{new}}(\\Gamma_0(N))$ and $g \\in S_2^{\\mathrm{new}}(\\Gamma_0(M))$ contained in the LMFDB (i.e., with $N, M \\leq 10\\,000$) and with coefficient fields of degree $\\leq 4$. Passing from "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.19813","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.19813/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}