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We prove combinatorial formulas for the dimension $\\gamma_{\\mathcal{M}_{\\pi}}(m)$ of the automorphism group scheme of $\\mathcal{M}_{\\pi}$ at finite level $m$ and the number of connected components of the endomorphism group scheme of $\\mathcal{M}_{\\pi}$ at finite level $m$. As an application, we show that if $\\mathcal{M}_{\\pi}$ is a nonordinary Dieudonn\\'e module defined by a cycle $\\pi$, then $\\gamma_{\\mathcal{M}_{\\pi}}"},"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":"1812.03577","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-09T23:19:27Z","cross_cats_sorted":[],"title_canon_sha256":"b8f9e249b105fca275f3f03dea787bfa47b527f54a423d6f15f05ccb86816ff3","abstract_canon_sha256":"67b4c8b08dfaebcd6fc9dbcb5ea1207ccfca60b49b7ec43f2e52f9d46a72fcf3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:12.477538Z","signature_b64":"noua2Vv29/B2AtpKyXRq+suuvJFdkgSPxApkQbVbi65jOc7ZCGlJkcagjZtfoNWDYMJqFdeeHElJuPJHgkNaBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ead4e6962e25fd67316ea8784714993dae3f70b2737417ada295612ac9ef56c","last_reissued_at":"2026-05-17T23:41:12.476976Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:12.476976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dimensions of automorphism group schemes of finite level truncations of $F$-cyclic $F$-crystals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiao Xiao, Zeyu Ding","submitted_at":"2018-12-09T23:19:27Z","abstract_excerpt":"Let $\\mathcal{M}_{\\pi}$ be an $F$-cyclic $F$-crystal $\\mathcal{M}_{\\pi}$ over an algebraically closed field defined by a permutation $\\pi$ and a set of prescribed Hodge slopes. 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