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These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. For special values of \\lambda, with s=1 and r=3, our congruences are stronger than what can be predicted by the theory of formal groups because of the p"},"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":"1210.4489","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-16T16:57:33Z","cross_cats_sorted":[],"title_canon_sha256":"f98b8f55fbc830d39be7ed1622e1f2a395ba4fc57f1e5fb563450427c4d16413","abstract_canon_sha256":"119c6d9885e27d0dcdad8021a85685a3cf9e4d4a11b6777628e038a4e0b0246d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:25.198036Z","signature_b64":"LeoXCn6FIrh0IZRPH7aRM34t7dn5xiU9fAMSHqrsp23Pe0qg3TgYQJit47KcvEMpLuKorzpQe9HBBpLXFhIUCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9baa6a0c889ee35930b8f7b74bb3317b8abe25a9c7adf27fc4e6e9115bcfc78b","last_reissued_at":"2026-05-18T03:40:25.197424Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:25.197424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Supercongruences and Complex Multiplication","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Benjamin Sheller, Hao Yuan, Jonas Kibelbek, Kevin Moss, Ling Long","submitted_at":"2012-10-16T16:57:33Z","abstract_excerpt":"We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\\lambda)_{(mp^s-1)/2} = \\sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \\lambda^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. 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