{"paper":{"title":"A proof of $p$-adic Gross--Zagier theorem via BDP formula","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A wall-crossing argument using the BDP formula proves the p-adic Gross-Zagier theorem for both ordinary and non-ordinary cuspidal forms.","cross_cats":[],"primary_cat":"math.NT","authors_text":"K\\^az{\\i}m B\\\"uy\\\"ukboduk, Peter Neamti","submitted_at":"2026-04-15T13:24:06Z","abstract_excerpt":"This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \\geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This paper provides a new proof of the p-adic Gross--Zagier formula for the p-adic L-function associated with the base change of a normalised cuspidal eigen-newform f of weight k >= 2 (and families of such) to an imaginary quadratic field K, encompassing classical p-ordinary cases and non-ordinary scenarios including k > 2 and ord_p(a_p(f)) > 0.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The wall-crossing strategy centred on the BDP formula and the theory of Beilinson--Flach elements successfully extends to the non-ordinary cases and higher weights without requiring the traditional comparison of geometric and analytic kernels.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new proof of the p-adic Gross-Zagier formula is established via a wall-crossing strategy centered on the BDP formula and Beilinson-Flach elements, covering ordinary and non-ordinary cases for weight k >= 2 and families.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A wall-crossing argument using the BDP formula proves the p-adic Gross-Zagier theorem for both ordinary and non-ordinary cuspidal forms.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"576df29433925ba9796a67c1cedf585c4bc58879ac34691c5fd8420d15a30637"},"source":{"id":"2604.13854","kind":"arxiv","version":2},"verdict":{"id":"43405c5c-1040-4909-b4b3-9669720b5a44","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T12:38:06.704139Z","strongest_claim":"This paper provides a new proof of the p-adic Gross--Zagier formula for the p-adic L-function associated with the base change of a normalised cuspidal eigen-newform f of weight k >= 2 (and families of such) to an imaginary quadratic field K, encompassing classical p-ordinary cases and non-ordinary scenarios including k > 2 and ord_p(a_p(f)) > 0.","one_line_summary":"A new proof of the p-adic Gross-Zagier formula is established via a wall-crossing strategy centered on the BDP formula and Beilinson-Flach elements, covering ordinary and non-ordinary cases for weight k >= 2 and families.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The wall-crossing strategy centred on the BDP formula and the theory of Beilinson--Flach elements successfully extends to the non-ordinary cases and higher weights without requiring the traditional comparison of geometric and analytic kernels.","pith_extraction_headline":"A wall-crossing argument using the BDP formula proves the p-adic Gross-Zagier theorem for both ordinary and non-ordinary cuspidal forms."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.13854/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}