The cuspidalization of sections of arithmetic fundamental groups
Seminar Room 1, Newton Institute
All results presented in this talk are part of a joint work with Akio Tamagawa. We introduce the problem of cuspidalization of sections of arithmetic fundamental groups which relates the Grothendieck section conjecture 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.
We prove that good sections of arithmetic fundamental groups of hyperbolic curves can be lifted to sections of the maximal cuspidally abelian Galois group of the function field of the curve (under quite general assumptions). As an application we prove a (geometrically pro-p) p-adic local version of the Grothendieck section conjecture under the assumption that the existence of sections of cuspidally (pro-p) abelian arithmetic fundamental groups implies the existence of tame points. We also prove, using cuspidalization techniques, that for a hyperbolic curve X over a p-adic local field and a set of points S of X which is dense in the p-adic topology every section of the arithmetic fundamental group of U=X\S arises from a rational point. As a corollary 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.