{"paper":{"title":"On the distribution of the of Frobenius elements on elliptic curves over function fields","license":"","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Amilcar Pacheco","submitted_at":"2002-11-20T11:23:51Z","abstract_excerpt":"Let $C$ be a smooth projective curve over $\\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $\\phi:\\mathcal{E}\\to C$ its minimal regular model. For each $P\\in C$ such that $E$ has good reduction at $P$, i.e., the fiber $\\mathcal{E}_P=\\phi^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $\\mathcal{E}_P$ over the residue field $\\kappa_P$ of $P$ are of the form $q_P^{1/2}e^{i\\theta_P},q_{P}e^{-i\\theta_P}$, where $q_P=q^{\\deg(P)}$ and $0\\le\\theta_P\\le\\pi$. The goal of this note is to determine given an integer $B\\ge 1$, $\\alpha,\\beta\\in[0,\\pi]$ the number of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0211315","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"}