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Liu says that if $T/p^n T$ is torsion semi-stable (resp. crystalline) of uniformly bounded Hodge-Tate weights for all $n \\geq 1$, then $T$ is also semi-stable (resp. crystalline). In this note, we show that we can relax the condition of \"uniformly bounded Hodge-Tate weights\" to an unbounded (log-)growth condition."},"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":"1905.08020","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-20T12:10:23Z","cross_cats_sorted":[],"title_canon_sha256":"04299d37fc6a9605660a4557a25aeb6a7db6ddb2afcd8baab6dd74d01c16ccc9","abstract_canon_sha256":"08618e492a148a1d1124d4e60f3ae30770a324598d9e54669679b48600e34d3e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:47.857236Z","signature_b64":"89nYhod/2sa5LYisPm+AgAPgqjWmzB3y0RsbwImoFhVggZpDkbtrJJmrWMrpLjxx0NCn5hXml59PRHIomdDLBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d4711515c58c432c13fb3d6f252b15d4ad93cc4f1562697b1635994cef08c41","last_reissued_at":"2026-05-17T23:45:47.856801Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:47.856801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Limit of torsion semi-stable Galois representations with unbounded weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hui Gao","submitted_at":"2019-05-20T12:10:23Z","abstract_excerpt":"Let $K$ be a complete discrete valuation field of characteristic $(0, p)$ with perfect residue field, and let $T$ be an integral $\\mathbb{Z}_p$-representation of $\\mathrm{Gal}(\\overline{K}/K)$. 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