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We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to the corresponding Hilbert modular form at the places above $p$ where $E$ has split multiplicative reduction. Under some mild restrictions on $p$ and the conductor of $E$ we deduce the exceptional zero conjecture in the strong form (i.e.\\ where the automorphic $p$-adic periods are replaced by the $\\cL$-invariants of $E$ defined in terms of Tate periods) from a"},"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":"1207.2289","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-10T10:16:35Z","cross_cats_sorted":[],"title_canon_sha256":"b92f2e779fe582822e2f897d0265b99001718d7f0867c18de2614991c19c5241","abstract_canon_sha256":"f054dc0aa551be2331d8a6e5f52b7f23f3eff6f214ddaa0ff57d50d132ffa39b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:15.957685Z","signature_b64":"FkOzbR4xEtcO5xVmQmDxlWnQN5CQv7f4d18td5sbtZqZx4ymVtlkaOCJB3guZLoUledTpdgI+4ljc7MVuz/7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc385052c988dabbda9fc9960b30072ecda78930d9e9cf246faa6494be146879","last_reissued_at":"2026-05-18T03:36:15.957133Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:15.957133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On special zeros of $p$-adic $L$-functions of Hilbert modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael Spiess","submitted_at":"2012-07-10T10:16:35Z","abstract_excerpt":"Let $E$ be a modular elliptic curve over a totally real number field $F$. 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