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Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a fixed finite abelian group $A$. We revisit these formulas and determine a simple expression for the zeta function $Z_p(A; t)$, the generating function for these $m$th power moments. In particular, we find that \\[ Z_p(A;t) = \\frac{\\widehat{Z}_p(A; t)}{\\displaystyle \\prod_{a \\in \\textrm{"},"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":"1807.00749","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-07-02T15:38:45Z","cross_cats_sorted":[],"title_canon_sha256":"fa42a431700e95eecbdf770fcfa2e4cd37eede312394befa266b9b81ad740123","abstract_canon_sha256":"41890886873b8b5fa8e88791bdc5fe39c53f9342ade6b02e61f13c5b3dfe66d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:51.536013Z","signature_b64":"HjWNdrbFYm+Opi8d8yvVdwn0MS9uN3anKwMr71JRg97OT/VgfPn4JYlvGLO0J+yB/KhlvQHp3D9vdVpwPbbwAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8453b071493c975047d78470d0b947474308085ee45c8e1a16285ec68433f171","last_reissued_at":"2026-05-18T00:11:51.535351Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:51.535351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generating functions for power moments of elliptic curves over $\\mathbb{F}_p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Katharine Woo, Katherine Gallagher, Katja Vassilev, Lucia Li, Naomi Sweeting","submitted_at":"2018-07-02T15:38:45Z","abstract_excerpt":"Seminal works by Birch and Ihara gave formulas for the $m$th power moments of the traces of Frobenius endomorphisms of elliptic curves over $\\mathbb{F}_{p}$ for primes $p \\geq 5$. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a fixed finite abelian group $A$. We revisit these formulas and determine a simple expression for the zeta function $Z_p(A; t)$, the generating function for these $m$th power moments. 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