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Two key ingredients in the argument are an extension to this setting of a $p$-adic formula of Bertolini-Darmon-Prasanna obtained in our earlier work, and an exceptional zero formula for Heegner points.\n  By independent approaches different from ours, a similar $p$-converse theorem was obtained by Skinner--Zhang under additional ramification hypotheses on $E[p]$, and by Venerucci assuming finiteness of the $p$-primary part of the Tate-Shafarevich g"},"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":"2409.01360","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-09-02T16:24:43Z","cross_cats_sorted":[],"title_canon_sha256":"2eae7af9c26b2ab708cc244e000406aaa04cef73cff152fbc5d3c4727b6e05b6","abstract_canon_sha256":"3259d93aed7380fdbf36ef9abba0b6a38a4df043febeb71b80647217b1512ebd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:02:28.173565Z","signature_b64":"LIvISgjuhBbOs2rmRIRfY3V11knmGgVezbrSikOU4FIGz+gUwprD8uZj+GtS3C0i5kbq2tS7/P4dymiRjZvYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8e266994efded37157513e4b017b73c6b327aa890293709e5e7add4f20eeac5","last_reissued_at":"2026-07-05T09:02:28.173142Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:02:28.173142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exceptional zeros for Heegner points and $p$-converse to the theorem of Gross-Zagier and Kolyvagin","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Francesc Castella","submitted_at":"2024-09-02T16:24:43Z","abstract_excerpt":"We prove a $p$-converse to the theorem of Gross-Zagier and Kolyvagin for elliptic curves $E/\\mathbf{Q}$ at primes $p>3$ of multiplicative reduction. 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