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We then provide the way to compute $-d$ and $d^{-1}$ using arithmetic in $\\mathbb{Z}_{p}$ and $\\mathbb{Z}_{p^{m}}$, and characterize invertible elements in $E_{p,p^m}$. Moreover, we introduce the minimal polynomial for each element in $E_{p,p^m}$ and given its applications."},"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":"1605.00805","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-03T09:27:51Z","cross_cats_sorted":[],"title_canon_sha256":"4e1cc51dfe3c9591666308bcc6653d9431999e85a7b6adaae4599b391cdcefe3","abstract_canon_sha256":"bd14e494c3edd0b98efc9afc15d3dab756c10efe3ca328d44ad4d2035b0356c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:47.997631Z","signature_b64":"IJOtgMTMI0+tp1MVt7/9poejvqmA6E/VK2shlT/qXjTcc1jhpHFNQAFWaUt87h0FW7bM8lPPrUayTV4VjCYGCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3a80f1f1495a7f5632ab289f35f6abf309582e8579584e5db06b6af71a0a76a","last_reissued_at":"2026-05-18T01:15:47.997095Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:47.997095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the arithmetic of the endomorphism ring End($\\mathbb{Z}_{p}\\times\\mathbb{Z}_{p^{m}}$)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hualu Liu, Xiusheng Liu","submitted_at":"2016-05-03T09:27:51Z","abstract_excerpt":"For a prime $p$, let $E_{p,p^m}=\\{\\begin{pmatrix}a&b\\\\p^{m-1}c&d\\end{pmatrix}|a,b,c\\in\\mathbb{Z}_{p},~\\mathrm{and}~d\\in \\mathbb{Z}_{p^{m}}\\}$. We first establish a ring isomorphism from $\\mathrm{End}(\\mathbb{Z}_p\\times\\mathbb{Z}_p^m)$ onto $E_{p,p^m}$. We then provide the way to compute $-d$ and $d^{-1}$ using arithmetic in $\\mathbb{Z}_{p}$ and $\\mathbb{Z}_{p^{m}}$, and characterize invertible elements in $E_{p,p^m}$. 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