{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T02:49:10Z","timestamp":1776826150989,"version":"3.51.2"},"reference-count":7,"publisher":"American Mathematical Society (AMS)","issue":"213","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n S<\/mml:mi>\n S<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a surface in\n \n \n \n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u00d7\n \n <\/mml:mo>\n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n \\mathbb {P}^2\\times \\mathbb {P}^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n given by the intersection of a (1,1)-form and a (2,2)-form. Then\n \n \n \n S<\/mml:mi>\n S<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a K3 surface with two noncommuting involutions\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n x<\/mml:mi>\n <\/mml:msup>\n \\sigma ^x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n y<\/mml:mi>\n <\/mml:msup>\n \\sigma ^y<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In 1991 the second author constructed two height functions\n \n \n \n \n \n \n h<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msup>\n \\hat {h}^+<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \n h<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \u2212\n \n <\/mml:mo>\n <\/mml:msup>\n \\hat {h}^-<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which behave canonically with respect to\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n x<\/mml:mi>\n <\/mml:msup>\n \\sigma ^x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n y<\/mml:mi>\n <\/mml:msup>\n \\sigma ^y<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights\n \n \n \n \n \n \n \u2211\n \n <\/mml:mo>\n v<\/mml:mi>\n <\/mml:munder>\n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \u00b1\n \n <\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n \u22c5\n \n <\/mml:mo>\n \n ,<\/mml:mo>\n v<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\sum _v \\hat {\\lambda }^\\pm (\\,\\cdot \\,,v)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We discuss how the geometry of the surface\n \n \n \n S<\/mml:mi>\n S<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is related to formulas for the local heights, and we give practical algorithms for computing the involutions\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n x<\/mml:mi>\n <\/mml:msup>\n \\sigma ^x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n y<\/mml:mi>\n <\/mml:msup>\n \\sigma ^y<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the local heights\n \n \n \n \n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n \u22c5\n \n <\/mml:mo>\n \n ,<\/mml:mo>\n v<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\hat {\\lambda }^+(\\,\\cdot \\,,v)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \u2212\n \n <\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n \u22c5\n \n <\/mml:mo>\n \n ,<\/mml:mo>\n v<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\hat {\\lambda }^-(\\,\\cdot \\,,v)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and the canonical heights\n \n \n \n \n \n \n h<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msup>\n \\hat {h}^+<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n \n \n h<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \u2212\n \n <\/mml:mo>\n <\/mml:msup>\n \\hat {h}^-<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-96-00680-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"259-290","source":"Crossref","is-referenced-by-count":3,"title":["Computing the canonical height on K3 surfaces"],"prefix":"10.1090","volume":"65","author":[{"given":"Gregory","family":"Call","sequence":"first","affiliation":[]},{"given":"Joseph","family":"Silverman","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","unstructured":"G. Call, Geometry and heights on certain K3 surfaces, in preparation."},{"issue":"2","key":"2","first-page":"163","article-title":"Canonical heights on varieties with morphisms","volume":"89","author":"Call, Gregory S.","year":"1993","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1810-2","volume-title":"Fundamentals of Diophantine geometry","author":"Lang, Serge","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0387908374"},{"key":"4","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":"https:\/\/id.crossref.org\/isbn\/0387962034"},{"key":"5","doi-asserted-by":"crossref","unstructured":"J.H. Silverman, The arithmetic of elliptic curves. I, Graduate Texts in Math., vol. 106, vol. 51, Springer-Verlag, Berlin and New York, 1988, pp. 339\u2013358. 87g:11070","DOI":"10.1090\/S0025-5718-1988-0942161-4"},{"key":"6","unstructured":"J. Tate, Letter to J.-P. Serre, Oct. 1, 1979."},{"key":"7","doi-asserted-by":"publisher","first-page":"596","DOI":"10.1090\/S0002-9904-1939-07046-8","article-title":"A general continued fraction expansion","volume":"45","author":"Leighton, Walter","year":"1939","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00680-1\/S0025-5718-96-00680-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00680-1\/S0025-5718-96-00680-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:02:13Z","timestamp":1776718933000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00680-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":7,"journal-issue":{"issue":"213","published-print":{"date-parts":[[1996,1]]}},"alternative-id":["S0025-5718-96-00680-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00680-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1996]]}}}