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We prove that $\\Gamma_f$ radializes $f$ if and only if $\\Gamma_X$ controls the pushforward of the constant $p$-adic differential equation $f_*(\\mathcal{O}_Y,d_Y)$.\n  Furthermore, when $f$ is a finite \\'etale morphism of open unit discs, we prove that $f$ is radial if and only if the number of preimages of a point $x\\in X$, counte"},"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":"1901.07644","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-01-22T23:23:36Z","cross_cats_sorted":[],"title_canon_sha256":"ec2cff121a480d6875e0b2e4e4d2d7f35732de36240822170d3efd3fd479eea5","abstract_canon_sha256":"c9bd4eca24314f44394c708546db9e56c81285697f264474619d1cce9955368c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:41.968269Z","signature_b64":"ZlBNecp3zlphUMcUG5d8BmO6IyEBtojJwtOWc7z7jHy/BSVEjuam5UcdsDMzw71l0agujfUGMmb79ki8mvmtAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1d11832f3843f2ec34f42925cd97b9e9f6b11cef78d5cd159c8db53d88dbedb","last_reissued_at":"2026-05-17T23:55:41.967631Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:41.967631Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Metric uniformization of morphisms of Berkovich curves via $p$-adic differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Francesco Baldassarri, Velibor Bojkovi\\'c","submitted_at":"2019-01-22T23:23:36Z","abstract_excerpt":"We consider a finite \\'etale morphism $f:Y \\to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton $\\Gamma_f=(\\Gamma_Y,\\Gamma_X)$\n  of the morphism $f$. 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