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We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in the full eigencurve C^{full}(l_{1}l_{2}). Further, we show that, E_{2}^{crit_{p}}, is \\'etale over the weight space in the eigencurve C(l). As a consequence, we show that level lowering conjecture of Paulin fails to hold at E_{2}^{crit_{p},ord_{l}}."},"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":"1307.4846","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-18T07:13:08Z","cross_cats_sorted":[],"title_canon_sha256":"d1dc336c7efb7d79d18f6268fb09793d9e308a1f0e01c0209151f7a1e29937fa","abstract_canon_sha256":"379cc7f182f5c8a778479a259fbb850a6612508a8ed69d385e1f71b0e7468798"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:48.729446Z","signature_b64":"8YbNPT5vBqBOtJW4mNMtJA+081rRcXk1Uk2zayfg8rvnMrt/yK8NbWXTEjlKTfyYD9kri4+QOKd+51tVkLyIAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"461231fbe6ed5cac565c32a714363979d6f2bf20ab3903f092bf164bdbda3eba","last_reissued_at":"2026-05-18T02:39:48.728992Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:48.728992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometry of the eigencurve at critical Eisenstein series of weight 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dipramit Majumdar","submitted_at":"2013-07-18T07:13:08Z","abstract_excerpt":"In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. 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