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SUMMARY:Levent Alpoge (Columbia University)
DTSTART:20200820T173000Z
DTEND:20200820T183000Z
DTSTAMP:20260423T024539Z
UID:MAGIC/18
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MAGIC/18/">E
 ffectivity in Faltings' Theorem.</a>\nby Levent Alpoge (Columbia Universit
 y) as part of MAGIC (Michigan - Arithmetic Geometry Initiative - Columbia)
 \n\n\nAbstract\nIn joint work with Brian Lawrence we show that\, assuming\
 nstandard motivic conjectures (Fontaine-Mazur\, Grothendieck-Serre\,\nHodg
 e\, Tate)\, there is a finite-time algorithm that\, on input $(K\,C)$\nwit
 h $K$ a number field and $C/K$ a smooth projective hyperbolic curve\,\nout
 puts $C(K)$. On the other hand\, in certain cases (i.e. after\nrestricting
  the inputs $(K\,C)$ --- e.g. so that $K/\\mathbb Q$ is totally real and\n
 of odd degree) there is an unconditional finite-time algorithm to\ncompute
  $(K\,C)\\mapsto C(K)$\, using potential modularity theorems. I will\ndisc
 uss these two results\, focusing in the latter case on how to\nuncondition
 ally compute the $K$-rational points on the curves $C_a : x^6\n+ 4y^3 = a^
 2$ (i.e. $a\\in K^\\times$ fixed) when $K/\\mathbb Q$ is totally real of\n
 odd degree.\n\n(The talk will cover Chapters 7\, 9\, and 11 of my thesis\,
  available on\ne.g. my website.)\n
LOCATION:https://researchseminars.org/talk/MAGIC/18/
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