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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0211315","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2002-11-20T11:23:51Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"dff2e3c0c506140b671aac756624eb9b5042cf366cc06ac3d199a75146465be9","abstract_canon_sha256":"ad444560265d6696eac77b03a029960c4915a2212a38fb8f49ec3cf376d91300"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:29.325861Z","signature_b64":"IN7dv6RUR3jdxTXAaz/tjrguj+iGrvyB+VOm/ZVMHIt7Tqt/L/bPoCfSGQsz/TcA1JptOXxU+FmThxq/huS+BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc17c4330ba6821cc7810b95eb89f2957a71057e060d6817c07c21183274ff27","last_reissued_at":"2026-05-18T01:38:29.325238Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:29.325238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0211315","created_at":"2026-05-18T01:38:29.325342+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0211315v1","created_at":"2026-05-18T01:38:29.325342+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0211315","created_at":"2026-05-18T01:38:29.325342+00:00"},{"alias_kind":"pith_short_12","alias_value":"3QL4IMYLU2BB","created_at":"2026-05-18T12:25:50.845339+00:00"},{"alias_kind":"pith_short_16","alias_value":"3QL4IMYLU2BBZR4B","created_at":"2026-05-18T12:25:50.845339+00:00"},{"alias_kind":"pith_short_8","alias_value":"3QL4IMYL","created_at":"2026-05-18T12:25:50.845339+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV","json":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV.json","graph_json":"https://pith.science/api/pith-number/3QL4IMYLU2BBZR4BBOK6XCPSSV/graph.json","events_json":"https://pith.science/api/pith-number/3QL4IMYLU2BBZR4BBOK6XCPSSV/events.json","paper":"https://pith.science/paper/3QL4IMYL"},"agent_actions":{"view_html":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV","download_json":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV.json","view_paper":"https://pith.science/paper/3QL4IMYL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0211315&json=true","fetch_graph":"https://pith.science/api/pith-number/3QL4IMYLU2BBZR4BBOK6XCPSSV/graph.json","fetch_events":"https://pith.science/api/pith-number/3QL4IMYLU2BBZR4BBOK6XCPSSV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV/action/storage_attestation","attest_author":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV/action/author_attestation","sign_citation":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV/action/citation_signature","submit_replication":"https://pith.science/pith/3QL4IMYLU2BBZR4BBOK6XCPSSV/action/replication_record"}},"created_at":"2026-05-18T01:38:29.325342+00:00","updated_at":"2026-05-18T01:38:29.325342+00:00"}