On the ‘Section Conjecture’ in anabelian geometry

Abstract

Let X be a smooth projective curve of genus > 1 over a field K with function field K (X ), let π1(X ) be the arithmetic fundamental group of X over K and let GF denote the absolute Galois group of a field F . The section conjecture in Grothendieck’s anabelian geometry says that the sections of the canonical projection π1(X ) ↠ GK are (up to conjugation) in one-to-one correspondence with the K-rational points of X, if K is finitely generated over ℚ. The birational variant conjectures a similar correspondence w.r.t. the sections of the projection GK (X ) ↠ GK .