BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jason Bell (University of Waterloo) DTSTART:20210614T111500Z DTEND:20210614T121500Z DTSTAMP:20260423T023048Z UID:WarsawNT/35 DESCRIPTION:Title: Effective isotrivial Mordell-Lang in positive characteristic\nby Jas on Bell (University of Waterloo) as part of Warsaw Number Theory Seminar\n \nLecture held in currently online.\n\nAbstract\nThe Mordell-Lang conjectu re (now a theorem\, proved by Faltings\, Vojta\, McQuillan\, …) asserts that if $G$ is a semiabelian variety $G$ defined over an algebraically clo sed field of characteristic zero\, $X$ is a subvariety of $G$\, and $\\Gam ma$ is a finite rank subgroup of $G$\, then $X\\cap \\Gamma$ is a finite u nion of cosets of $\\Gamma$. In positive characteristic\, the naive trans lation of this theorem does not hold\, however Hrushovski\, using model th eoretic techniques\, showed that in some sense all counterexamples arise f rom semiabelian varieties defined over finite fields (the isotrivial case) . This was later refined by Moosa and Scanlon\, who showed in the isotriv ial case that the intersection of a subvariety of a semiabelian variety $G $ with a finitely generated subgroup $\\Gamma$ of $G$ that is invariant un der the Frobenius endomorphism $F:G\\to G$ is a finite union of sets of th e form $S+A$\, where $A$ is a subgroup of $\\Gamma$ and $S$ is a sum of or bits under the map $F$. We show how how one can use finite-state automat a to give a concrete description of these intersections $\\Gamma\\cap X$ i n the isotrivial setting\, by constructing a finite machine that identifie s all points in the intersection. In particular\, this allows us to give d ecision procedures for answering questions such as: is $X\\cap \\Gamma$ em pty? finite? does it contain a coset of an infinite subgroup? In addition\ , we are able to read off coarse asymptotic estimates for the number of po ints of height $\\le H$ in the intersection from the machine. This is joi nt work with Dragos Ghioca and Rahim Moosa.\n LOCATION:https://researchseminars.org/talk/WarsawNT/35/ END:VEVENT END:VCALENDAR