Finite descent obstruction and non-abelian reciprocity

Abstract: For a nice algebraic variety $X$ over a number field $F$, one of the central problems of Diophantine Geometry is to locate precisely the set $X(F)$ inside $X(\A_F)$, where $\A_F$ denotes the ring of ad`eles of $F$. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for (certain) finite \'etale group schemes over $F$ on $X$. More recently, Kim proposed an iterative construction of another subset of $X(\A_F)$ which contains the set of rational points. In this paper, we compare the two constructions. Our main result shows that the two approaches are essentially equivalent.