Pure Maths Colloquium: On Grothendieck's anabelian section conjecture for curves
24 November 2008 16:15 in CM 107
I will discuss some recent progress on Grothendieck's anabelian section conjecture obtained jointly with A. Tamagawa. The conjecture predicts that rational points of hyperbolic curves over number fields all arise from sections of the arithmetic fundamental group of the curve. We develop/introduce the theory of cuspidalization of sections of arithmetic fundamental groups. As a consequence we prove that the p-adic version of the anabelian section conjceture holds under a certain condition (roughly speaking the condition is that the existence of a section implies the existence of a tame point). We also prove that a section of the arithmetic fundamental group of
X\S, where X is a proper and smooth curve over a p-adic field and S is a set of points which are every where dense (in the p-adic topology) arise from a rational point. Finally, we prove that the existence of a section of the arithmetic absolute Galois group of a curve X over a number field implies that the set of adelic points of X is non empty.