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We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences modulo the integers $n_1, n_2, n_3, n_4$ on certain combinations of coe"},"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":"1307.0863","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-02T21:52:08Z","cross_cats_sorted":[],"title_canon_sha256":"2b1065a42283a1f20d94514cbda3b99d241840cfbb74d540d140c384a99fa604","abstract_canon_sha256":"dfab1d02cdb735c32cb33ecb7153460091cb9f8127cfd7f6c7db535c1fa727d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:23.831368Z","signature_b64":"smeZzZ0H3Now9ppw7rc5/sTUk9p9eDMZrscZaJspb91De43iFSylixfu7xhtef6sowKTml4fC4jcW3sVdJREBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0bb28d3560f9214f83ebda90a9cb7f79fba7ddfbf0d338e448b62cca3bedeee3","last_reissued_at":"2026-05-18T03:19:23.830555Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:23.830555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Abelian surfaces over finite fields with prescribed groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Arul Shankar, Chantal David, Derek Garton, Ethan Smith, Lola Thompson, Zachary Scherr","submitted_at":"2013-07-02T21:52:08Z","abstract_excerpt":"Let A be an abelian surface over F_q, the field of q elements. 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