{"paper":{"title":"Conditional algorithmic Mordell","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Brian Lawrence, Levent Alp\\\"oge","submitted_at":"2024-08-21T14:26:05Z","abstract_excerpt":"We specify a Turing machine $T_{\\text{Mordell}}$ with the following properties.\n  1. On input $(K,C/K)$, with $K/\\mathbb{Q}$ a number field and $C/K$ a smooth projective hyperbolic curve, if $T_{\\text{Mordell}}$ terminates, then it outputs $C(K)$.\n  2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that $T_{\\text{Mordell}}$ always terminates.\n  Similarly we specify a Turing machine $T_{\\text{Shafarevich}}$ with the following properties.\n  1. On input $(g, K,S, d)$, with $g, d \\in \\mathbb{Z}^+$, $K/\\mathbb{Q}$ a number field, and $S$ a finite set of places of $K$, if $T_{\\text{Shafarevic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.11653","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.11653/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"}