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We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the zero point of $J$). For each point $P=(a,b)\\in C(K)$ there are $2^{2g}$ points $\\frac{1}{2}P \\in J(K)$. We describe explicitly the Mumford represesentations of all $\\frac{1}{2}P$. The rationality question"},"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":"1606.05252","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-06-16T16:41:39Z","cross_cats_sorted":[],"title_canon_sha256":"c52dcc3e424c920be24650d31ad7b345fe1254bc2c6fdee9ff42afc67938bc23","abstract_canon_sha256":"92a96111ae8844353ba4df95dd84cbd57ad566d292fae89575692e9443cee5e7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:37.791789Z","signature_b64":"2/5sDHUFaiyDLPt87ut+/k41rYoU3eINBeUvFr2zRn2nMg2co3xtaARcw00gcuASBznG1bMRfo8Qf4T/gTOGAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"edb91b9b63bf8d928ea4429eb42d3fff79bf100bbfe6f1ed641cc8ec60c7f5f4","last_reissued_at":"2026-05-18T00:56:37.791019Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:37.791019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Division by 2 on hyperelliptic curves and jacobians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Yuri G. Zarhin","submitted_at":"2016-06-16T16:41:39Z","abstract_excerpt":"Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over $K$ and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the zero point of $J$). For each point $P=(a,b)\\in C(K)$ there are $2^{2g}$ points $\\frac{1}{2}P \\in J(K)$. We describe explicitly the Mumford represesentations of all $\\frac{1}{2}P$. 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