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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"},"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":"1906.09407","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-22T07:55:22Z","cross_cats_sorted":[],"title_canon_sha256":"741656181ba05436910d5410d53fd31f0e6455354d6921f7dff3d9f6a638a526","abstract_canon_sha256":"fb886cbe48a029e2417cdc899c4826c6e225a3cacb31a920c0d250a032bbe6ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:38.221370Z","signature_b64":"6jgrp1L1Mxsxv/sbH5B2VguEyMEBimeCV7lxuaI5E1kxxLMSv4VnHHGzayC3wjoVkGmwmBOFtiqQpsakUbZQAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b01680e89a2c0c3a9369fdddbc5d229a27f4dd136b2d1811c68f1745a1c0cffd","last_reissued_at":"2026-05-17T23:42:38.220611Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:38.220611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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