Galois sections in absolute anabelian geometry

Abstract

We show that isomorphisms between arithmetic fundamental groups of hyperbolic curves over $p$-adic local fields preserve the decomposition groups of all closed points (respectively, closed points arising from torsion points of the underlying elliptic curve), whenever the hyperbolic curves in question are isogenous to hyperbolic curves of genus zero defined over a number field (respectively, are once-punctured elliptic curves [which are not necessarily defined over a number field]). We also show that, under certain conditions, such isomorphisms preserve certain canonical ``integral structures'' at the cusps [i.e., points at infinity] of the hyperbolic curve.