{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,9]],"date-time":"2024-06-09T22:54:42Z","timestamp":1717973682392},"reference-count":15,"publisher":"Wiley","license":[{"start":{"date-parts":[[2012,9,1]],"date-time":"2012-09-01T00:00:00Z","timestamp":1346457600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove that under any projective embedding of an abelian variety <jats:italic>A<\/jats:italic> of dimension <jats:italic>g<\/jats:italic>, a complete set of addition laws has cardinality at least <jats:italic>g<\/jats:italic>+1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in \u2119<jats:sup>2<\/jats:sup>. In contrast, we prove, moreover, that if <jats:italic>k<\/jats:italic> is any field with infinite absolute Galois group, then there exists for every abelian variety <jats:italic>A<\/jats:italic>\/<jats:italic>k<\/jats:italic> a projective embedding and an addition law defined for every pair of <jats:italic>k<\/jats:italic>-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or the embedding in \u2119<jats:sup>15<\/jats:sup>, respectively, up to a finite number of counterexamples for \u2223<jats:italic>k<\/jats:italic>\u2223\u22645 .<\/jats:p>","DOI":"10.1112\/s1461157012001027","type":"journal-article","created":{"date-parts":[[2012,11,1]],"date-time":"2012-11-01T08:23:51Z","timestamp":1351758231000},"page":"308-316","source":"Crossref","is-referenced-by-count":3,"title":["Complete addition laws on abelian varieties"],"prefix":"10.1112","volume":"15","author":[{"given":"Christophe","family":"Arene","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Kohel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christophe","family":"Ritzenthaler","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2012,9,1]]},"reference":[{"key":"S1461157012001027_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnt.2010.12.001"},{"key":"S1461157012001027_ref1","volume-title":"Arithmetic, geometry, cryptography and coding theory 2011","author":"Arene","year":"2012"},{"key":"S1461157012001027_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1810-2"},{"key":"S1461157012001027_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-3810-9"},{"key":"S1461157012001027_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(87)90077-9"},{"key":"S1461157012001027_ref4","unstructured":"[4] Bernstein D.\u00a0J. and Lange T. , \u2018Complete addition laws for all elliptic curves over finite fields\u2019, presentation, 2009, cr.yp.to\/talks\/2009.07.17\/slides.pdf."},{"key":"S1461157012001027_ref14","volume-title":"Curves and their Jacobians","author":"Mumford","year":"1975"},{"key":"S1461157012001027_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-76900-2_3"},{"key":"S1461157012001027_ref13","first-page":"29","volume-title":"Questions on algebraic varieties (C.I.M.E. summer school III, Ciclo, Varenna, 1969)","author":"Mumford","year":"1970"},{"key":"S1461157012001027_ref6","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.1995.1088"},{"key":"S1461157012001027_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9930-8"},{"key":"S1461157012001027_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01388526"},{"key":"S1461157012001027_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-06307-1"},{"key":"S1461157012001027_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8534-7"},{"key":"S1461157012001027_ref2","unstructured":"[2] Bernstein D.\u00a0J. , Kohel D. and Lange T. , \u2018Twisted hessian curves\u2019, Preprint, 2009."}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157012001027","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,6]],"date-time":"2019-06-06T20:47:22Z","timestamp":1559854042000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157012001027\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,1]]},"references-count":15,"alternative-id":["S1461157012001027"],"URL":"https:\/\/doi.org\/10.1112\/s1461157012001027","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,9,1]]}}}