{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T17:46:15Z","timestamp":1648662375324},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2002,3,7]],"date-time":"2002-03-07T00:00:00Z","timestamp":1015459200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Q Subscript p\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mi>p<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Q}_p<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We illustrate the practicality of the method with some examples of hyperelliptic curves of genus <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n  <mml:semantics>\n    <mml:mn>1<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>.<\/p>","DOI":"10.1090\/s0025-5718-01-01275-3","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1661-1674","source":"Crossref","is-referenced-by-count":1,"title":["Sieving for rational points on hyperelliptic curves"],"prefix":"10.1090","volume":"70","author":[{"given":"Samir","family":"Siksek","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,3,7]]},"reference":[{"key":"1","series-title":"Addison-Wesley Series in Computer Science and Information Processing","isbn-type":"print","volume-title":"Data structures and algorithms","author":"Aho, Alfred V.","year":"1983","ISBN":"http:\/\/id.crossref.org\/isbn\/0201000237"},{"key":"2","series-title":"London Mathematical Society Student Texts","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139171885","volume-title":"Local fields","volume":"3","author":"Cassels, J. W. S.","year":"1986","ISBN":"http:\/\/id.crossref.org\/isbn\/0521304849"},{"key":"3","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1112\/jlms\/s1-41.1.193","article-title":"Diophantine equations with special reference to elliptic curves","volume":"41","author":"Cassels, J. W. S.","year":"1966","journal-title":"J. London Math. Soc.","ISSN":"http:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1515\/crll.1998.001","article-title":"Second descents for elliptic curves","volume":"494","author":"Cassels, J. W. S.","year":"1998","journal-title":"J. Reine Angew. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"5","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02945-9","volume-title":"A course in computational algebraic number theory","volume":"138","author":"Cohen, Henri","year":"1993","ISBN":"http:\/\/id.crossref.org\/isbn\/3540556400"},{"key":"6","isbn-type":"print","volume-title":"Algebra. Vol. 1","author":"Cohn, P. M.","year":"1982","ISBN":"http:\/\/id.crossref.org\/isbn\/0471101699","edition":"2"},{"key":"7","isbn-type":"print","volume-title":"Algorithms for modular elliptic curves","author":"Cremona, J. E.","year":"1997","ISBN":"http:\/\/id.crossref.org\/isbn\/0521598206","edition":"2"},{"key":"8","unstructured":"[Cre2] J. E. Cremona, Personal Communication, 1996."},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"385","DOI":"10.4064\/aa-77-4-385-404","article-title":"Explicit 4-descents on an elliptic curve","volume":"77","author":"Merriman, J. R.","year":"1996","journal-title":"Acta Arith.","ISSN":"http:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"10","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1920-8","volume-title":"The arithmetic of elliptic curves","volume":"106","author":"Silverman, Joseph H.","year":"1986","ISBN":"http:\/\/id.crossref.org\/isbn\/0387962034"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01275-3\/S0025-5718-01-01275-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01275-3\/S0025-5718-01-01275-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:38:36Z","timestamp":1627605516000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01275-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,3,7]]},"references-count":10,"journal-issue":{"issue":"236","published-print":{"date-parts":[[2001,10]]}},"alternative-id":["S0025-5718-01-01275-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01275-3","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,3,7]]}}}