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The catch is that we cannot say whether\n  \\rho|_{G_p=Gal(\\bar{Q_p}/Q_p)\n  is crystalline or even potentially semistable. The second construction assumes the Generalized Riemann Hypothesis (GRH). With this assumption we can further arrange that \\rho|_{G_p} is crystalline at p. We remark that infin"},"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":"math/0003241","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2000-03-01T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"680770cb1e2dca9f44face02f98b94b7106372296e0bbb6b1b78d7c2d4b16ab2","abstract_canon_sha256":"76d553666d7b8f2e9e75850148beff8509b0d64502146caeb650706fd88ad58b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:38.685427Z","signature_b64":"cPS+WjC+rJWJxYH7LXBLQz4Zb2vvF8KcWck4e79OVtFYw7MLWXWjfGt8qchH7vXr69elR4Mtgj3flQ1A7XRZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"579edf58363713f4620b7891335d2ac25f0dac97d14c42d6cd4825608d89f58a","last_reissued_at":"2026-05-18T01:05:38.684877Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:38.684877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Infinitely ramified Galois representations","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ravi Ramakrishna","submitted_at":"2000-03-01T00:00:00Z","abstract_excerpt":"In this paper we show how to construct, for most p >= 5, two types of surjective representations\n  \\rho:G_Q=Gal(\\bar{Q}/Q) -> GL_2(Z_p)\n  that are ramified at an infinite number of primes. The image of inertia at almost all of these primes will be torsion-free. The first construction is unconditional. The catch is that we cannot say whether\n  \\rho|_{G_p=Gal(\\bar{Q_p}/Q_p)\n  is crystalline or even potentially semistable. The second construction assumes the Generalized Riemann Hypothesis (GRH). With this assumption we can further arrange that \\rho|_{G_p} is crystalline at p. 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