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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:The cuspidalization of sections of arithmetic fund
amental groups - Saidi\, M (Exeter)
DTSTART;TZID=Europe/London:20090730T113000
DTEND;TZID=Europe/London:20090730T123000
UID:TALK19269AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/19269
DESCRIPTION:All results presented in this talk are part of a j
oint work with Akio Tamagawa. We introduce the pro
blem of cuspidalization of sections of arithmetic
fundamental groups which relates the Grothendieck
section conjecture to its birational analog. We ex
hibit a necessary condition for a section of the a
rithmetic fundamental group of a hyperbolic curve
to arise from a rational point which we call the g
oodness condition.\nWe prove that good sections of
arithmetic fundamental groups of hyperbolic curve
s can be lifted to sections of the maximal cuspida
lly abelian Galois group of the function field of
the curve (under quite general assumptions). As an
application we prove a (geometrically pro-p) p-ad
ic local version of the Grothendieck section conje
cture under the assumption that the existence of s
ections of cuspidally (pro-p) abelian arithmetic f
undamental groups implies the existence of tame po
ints. We also prove\, using cuspidalization techni
ques\, that for a hyperbolic curve X over a p-adic
local field and a set of points S of X which is d
ense in the p-adic topology every section of the a
rithmetic fundamental group of U=XS arises from a
rational point. As a corollary we deduce that the
existence of a section of the absolute Galois grou
p of a function field of a curve over a number fie
ld implies that the set of adelic points of the cu
rve is non-empty.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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