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SUMMARY:The cuspidalization of sections of arithmetic fundamental groups -
  Saidi\, M (Exeter)
DTSTART:20090730T103000Z
DTEND:20090730T113000Z
UID:TALK19269@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:All results presented in this talk are part of a joint work wi
 th Akio Tamagawa. We introduce the problem of cuspidalization of sections 
 of arithmetic fundamental groups which relates the Grothendieck section co
 njecture to its birational analog. We exhibit a necessary condition for a 
 section of the arithmetic fundamental group of a hyperbolic curve to arise
  from a rational point which we call the goodness condition.\nWe prove tha
 t good sections of arithmetic fundamental groups of hyperbolic curves can 
 be lifted to sections of the maximal cuspidally abelian Galois group of th
 e function field of the curve (under quite general assumptions). As an app
 lication we prove a (geometrically pro-p) p-adic local version of the Grot
 hendieck section conjecture under the assumption that the existence of sec
 tions of cuspidally (pro-p) abelian arithmetic fundamental groups implies 
 the existence of tame points. We also prove\, using cuspidalization techni
 ques\, that for a hyperbolic curve X over a p-adic local field and a set o
 f points S of X which is dense in the p-adic topology every section of the
  arithmetic fundamental group of U=XS arises from a rational point. As a c
 orollary we deduce that the existence of a section of the absolute Galois 
 group of a function field of a curve over a number field implies that the 
 set of adelic points of the curve is non-empty.
LOCATION:Seminar Room 1\, Newton Institute
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