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SUMMARY:David Urbanik (Toronto)
DTSTART:20220310T220000Z
DTEND:20220310T230000Z
DTSTAMP:20260423T022806Z
UID:UCSD_NTS/59
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCSD_NTS/59/
 ">Effective Methods for Shafarevich Problems</a>\nby David Urbanik (Toront
 o) as part of UCSD number theory seminar\n\nLecture held in APM 7321.\n\nA
 bstract\nGiven a smooth projective family $f : X \\to S$ defined over\nthe
  ring of integers of a number field\, the Shafarevich problem is to\ndescr
 ibe those fibres of f which have everywhere good reduction. This\ncan be i
 nterpreted as asking for the dimension of the Zariski closure\nof the set 
 of integral points of $S$\, and is ultimately a difficult\ndiophantine que
 stion about which little is known as soon as the\ndimension of $S$ is at l
 east 2. Recently\, Lawrence and Venkatesh gave a\ngeneral strategy for add
 ressing such problems which requires as input\nlower bounds on the monodro
 my of f over essentially arbitrary closed\nsubvarieties of $S$. In this ta
 lk we review their ideas\, and describe\nrecent work which gives a fully e
 ffective method for computing these\nlower bounds. This gives a fully effe
 ctive strategy for solving\nShafarevich-type problems for essentially arbi
 trary families $f$.\n\nThis week's talk is in APM 7321 rather than APM 640
 2.\n\npre-talk at 1:20 pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/59/
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