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SUMMARY:Alex Best (VU Amsterdam)
DTSTART:20220427T150000Z
DTEND:20220427T160000Z
DTSTAMP:20260418T064544Z
UID:LNTS/67
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/67/">Th
 e S-unit equation and non-abelian Chabauty in depth 2</a>\nby Alex Best (V
 U Amsterdam) as part of London number theory seminar\n\nLecture held in Bu
 sh House S2.03\, King's College London.\n\nAbstract\nThe S-unit equation i
 s a classical and well-studied Diophantine equation\, with numerous connec
 tions to other Diophantine problems.\nRecent work of Kim and refinements d
 ue to Betts-Dogra have suggested new cohomological strategies to find rati
 onal and integral points on curves\, based on but massively extending the 
 classical method of Chabauty. At present\, these methods are only conjectu
 rally guaranteed to succeed in general\, but they promise several applicat
 ions in arithmetic geometry if they could be proved to always work.\nIn or
 der to better understand the conjectures of Kim that suggest that this met
 hod should work\, we consider the case of the thrice punctured projective 
 line\, in "depth 2"\, the "smallest" non-trivial extension of the classica
 l method. In doing so we get very explicit results for some S-unit equatio
 ns\, demonstrating the usability of the aforementioned cohomological metho
 ds in this setting. To do this we determine explicitly equations for (maps
  between) the (refined) Selmer schemes defined by Kim\, and Betts-Dogra\, 
 which turn out to have some particularly simple forms.\nThis is joint work
  with Alexander Betts\, Theresa Kumpitsch\, Martin Lüdtke\, Angus McAndre
 w\, Lie Qian\, Elie Studnia\, and Yujie Xu .\n
LOCATION:https://researchseminars.org/talk/LNTS/67/
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