{"paper":{"title":"A graph-theoretic approach to computing Selmer groups of elliptic curves $y^2 = x^3 + bx$ over $\\mathbb{Q}(i)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anthony Kling, Ben Savoie","submitted_at":"2024-10-30T05:50:04Z","abstract_excerpt":"We develop a graph-theoretic algorithm to compute the $\\varphi$-Selmer group of the elliptic curve $E_b: y^2 = x^3 + bx$ over $\\mathbb{Q}(i)$, where $b \\in \\mathbb{Z}[i]$ and $\\varphi$ is a degree 2 isogeny of $E_b$. We associate to $E_b$ a weighted graph $G_b$, whose vertices are the odd Gaussian primes dividing $b$, and whose edge weights are determined by the quartic residue symbol between pairs of these primes. By applying our algorithm, we explicitly compute the $\\varphi$-Selmer group of $E_b$ when $b$ is a product of inert primes, and we construct several infinite families of elliptic cu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.22714","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/2410.22714/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"}