{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T21:10:02Z","timestamp":1718399402625},"reference-count":32,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2016,6]]},"abstract":"<jats:p>We give a reduction of the function field Mordell\u2013Lang conjecture to the function field Manin\u2013Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the \u201cTheorem of the Kernel\u201d, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: \u201cquantifier-elimination\u201d for the corresponding [Formula: see text] where [Formula: see text] is a saturated separably closed field.<\/jats:p>","DOI":"10.1142\/s021906131650001x","type":"journal-article","created":{"date-parts":[[2016,3,7]],"date-time":"2016-03-07T04:05:51Z","timestamp":1457323551000},"page":"1650001","source":"Crossref","is-referenced-by-count":5,"title":["On function field Mordell\u2013Lang and Manin\u2013Mumford"],"prefix":"10.1142","volume":"16","author":[{"given":"Franck","family":"Benoist","sequence":"first","affiliation":[{"name":"Laboratoire de Math\u00e9matiques d\u2019Orsay, Universit\u00e9 Paris-Sud, CNRS, Universit\u00e9 Paris-Saclay, 91405 Orsay, France"}]},{"given":"Elisabeth","family":"Bouscaren","sequence":"additional","affiliation":[{"name":"Laboratoire de Math\u00e9matiques d\u2019Orsay, Universit\u00e9 Paris-Sud, CNRS, Universit\u00e9 Paris-Saclay, 91405 Orsay, France"}]},{"given":"Anand","family":"Pillay","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Notre Dame, 281 Hurley Hall, Notre Dame, IN 46556, USA"}]}],"member":"219","published-online":{"date-parts":[[2016,7,4]]},"reference":[{"key":"S021906131650001XBIB001","doi-asserted-by":"publisher","DOI":"10.1155\/S1073792892000126"},{"key":"S021906131650001XBIB002","doi-asserted-by":"publisher","DOI":"10.1017\/S147474801400022X"},{"key":"S021906131650001XBIB003","doi-asserted-by":"publisher","DOI":"10.1017\/S1474748008000145"},{"key":"S021906131650001XBIB004","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-09-00653-5"},{"key":"S021906131650001XBIB005","series-title":"Lecture Notes in Mathematics","volume-title":"Model Theory and Algebraic Geometry","volume":"1696","author":"Bouscaren E.","year":"1999","edition":"2"},{"key":"S021906131650001XBIB006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-01-02886-0"},{"key":"S021906131650001XBIB007","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1190150042"},{"key":"S021906131650001XBIB008","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0087235","volume-title":"Differential Algebraic Groups of Finite Dimension","volume":"1506","author":"Buium A.","year":"1992"},{"key":"S021906131650001XBIB009","doi-asserted-by":"publisher","DOI":"10.2307\/2946600"},{"key":"S021906131650001XBIB012","series-title":"Lecture Notes in Mathematics","volume-title":"Model Theory and Algebraic Geometry","volume":"1696","author":"Delon F.","year":"1999","edition":"2"},{"key":"S021906131650001XBIB013","doi-asserted-by":"publisher","DOI":"10.1007\/BF01394276"},{"key":"S021906131650001XBIB014","series-title":"Lecture Notes in Mathematics","volume-title":"Model Theory and Algebraic Geometry","volume":"1696","author":"Hindry M.","year":"1999","edition":"2"},{"key":"S021906131650001XBIB015","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-96-00202-0"},{"key":"S021906131650001XBIB016","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(01)00096-3"},{"key":"S021906131650001XBIB017","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-96-00180-4"},{"key":"S021906131650001XBIB018","series-title":"Lecture Notes in Mathematics","volume-title":"Model Theory and Algebraic Geometry","volume":"1696","author":"Lascar D.","year":"1999","edition":"2"},{"key":"S021906131650001XBIB019","series-title":"Lecture Notes in Logic","volume-title":"Model Theory of Fields","volume":"5","author":"Marker D.","year":"2003","edition":"2"},{"issue":"3","key":"S021906131650001XBIB020","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1215\/ijm\/1255985739","volume":"41","author":"Marker D.","year":"1997","journal-title":"Illinois J. 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