{"paper":{"title":"Rank and Bias in Families of Hyperelliptic Curves via Nagao's Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"\\'Alvaro Lozano-Robledo, Benjamin Logsdon, Seoyoung Kim, Steven J. Miller, Trajan Hammonds","submitted_at":"2019-06-22T07:55:22Z","abstract_excerpt":"Let $\\mathcal{X} : y^2 = f(x)$ be a hyperelliptic curve over $\\mathbb{Q}(T)$ of genus $g\\geq 1$. Assume that the jacobian of $\\mathcal{X}$ over $\\mathbb{Q}(T)$ has no subvariety defined over $\\mathbb{Q}$. Denote by $\\mathcal{X}_t$ the specialization of $\\mathcal{X}$ to an integer $T=t$, let $a_{\\mathcal{X}_t}(p)$ be its trace of Frobenius, and $A_{\\mathcal{X},r}(p) = \\frac{1}{p}\\sum_{t=1}^p a_{\\mathcal{X}_t}(p)^r$ its $r$-th moment. The first moment is related to the rank of the jacobian $J_\\mathcal{X}\\left(\\mathbb{Q}(T)\\right)$ by a generalization of a conjecture of Nagao: $$\\lim_{X \\to \\inft"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.09407","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"}