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In this paper, we study the $q$-adic valuations of the reciprocal roots in $\\mathbb{C}_p$ of $L$-functions associated to characters of the Galois group of $\\mathcal{P}$. We show that for all covers $\\mathcal{P}$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ for $n$ large, the valuations of these reciprocal "},"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":"1701.08733","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-30T18:05:11Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"5e18e43af80b3a134352d45215e0e8da2355f431a102498c66f96a532a47bbf2","abstract_canon_sha256":"2ddc477f4ab8a085ae6cb4cf4f1018447660b08f0baf0777416ea115be5e6a4a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:04.733910Z","signature_b64":"bRNuVE/Sna9ypwMJneGhR/7lxYwdSvm089dq7x/lPue+JoN7S8Av84aEPCtMNvp8mXzDHwsbWWUE4YgFNUZYDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49006f7ea98bd59d034ff38253e3a6b74430a1422ac84a118bd3b65f1fb586f5","last_reissued_at":"2026-05-18T00:35:04.733194Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:04.733194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On slopes of $L$-functions of $\\mathbb{Z}_p$-covers over the projective line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Hui June Zhu, Michiel Kosters","submitted_at":"2017-01-30T18:05:11Z","abstract_excerpt":"Let $\\mathcal{P}: \\cdots \\rightarrow C_2\\rightarrow C_1\\rightarrow {\\mathbb P}^1$ be a $\\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point, and is unramified at other points. 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