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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"},"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":"2408.11653","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-08-21T14:26:05Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"f6d0d82443d9aeb5325bd2e4b7c09902656fb0b7ded99e6abcc5134701b8ce7e","abstract_canon_sha256":"5db4bf66c3b1438fe4b39362d42f24865e6a471000bdc52428a94a57887700c8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:57:46.742879Z","signature_b64":"GvWOqrJBO/Mh63SeX+8VpIWYOj53daNMGWNDvGFJNXm+Ni3P56VTfhOj7XWD1bHW5hYd1TOS73SiiOROWMiwBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47b8ca9bb8f25e507ef5950fe5323a4dfee915975828af06be52b192f7f9c8cd","last_reissued_at":"2026-07-05T08:57:46.742407Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:57:46.742407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2408.11653","created_at":"2026-07-05T08:57:46.742466+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.11653v1","created_at":"2026-07-05T08:57:46.742466+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.11653","created_at":"2026-07-05T08:57:46.742466+00:00"},{"alias_kind":"pith_short_12","alias_value":"I64MVG5Y6JPF","created_at":"2026-07-05T08:57:46.742466+00:00"},{"alias_kind":"pith_short_16","alias_value":"I64MVG5Y6JPFA7XV","created_at":"2026-07-05T08:57:46.742466+00:00"},{"alias_kind":"pith_short_8","alias_value":"I64MVG5Y","created_at":"2026-07-05T08:57:46.742466+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.09512","citing_title":"Abelian surfaces of small conductor from genus 3 double covers","ref_index":1,"is_internal_anchor":false},{"citing_arxiv_id":"2605.16985","citing_title":"On Variable-Bounded Non-Linear Expansions of Presburger Arithmetic","ref_index":20,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX","json":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX.json","graph_json":"https://pith.science/api/pith-number/I64MVG5Y6JPFA7XVSUH6KMR2JX/graph.json","events_json":"https://pith.science/api/pith-number/I64MVG5Y6JPFA7XVSUH6KMR2JX/events.json","paper":"https://pith.science/paper/I64MVG5Y"},"agent_actions":{"view_html":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX","download_json":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX.json","view_paper":"https://pith.science/paper/I64MVG5Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.11653&json=true","fetch_graph":"https://pith.science/api/pith-number/I64MVG5Y6JPFA7XVSUH6KMR2JX/graph.json","fetch_events":"https://pith.science/api/pith-number/I64MVG5Y6JPFA7XVSUH6KMR2JX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX/action/storage_attestation","attest_author":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX/action/author_attestation","sign_citation":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX/action/citation_signature","submit_replication":"https://pith.science/pith/I64MVG5Y6JPFA7XVSUH6KMR2JX/action/replication_record"}},"created_at":"2026-07-05T08:57:46.742466+00:00","updated_at":"2026-07-05T08:57:46.742466+00:00"}