On the birational anabelian section conjecture

TL;DR

This paper explores the birational anabelian section conjecture over number fields, reducing it to elliptic curves and establishing conditions under which sections correspond to rational points, assuming finiteness of the Shafarevich-Tate group.

Contribution

It reduces the birational section conjecture to elliptic curves and links sections to rational points via a result of Stoll, under the assumption of Shafarevich-Tate group finiteness.

Findings

Reduction of the conjecture to elliptic curves.

Equivalence of sections arising from rational points.

Existence of double coverings satisfying the conjecture.

Abstract

Assuming the finiteness of the Shafarevich-Tate group of elliptic curves over number fields we make several observations on the birational Grotendieck anabelian setion conjecture. We prove that the birational setion conjecture for curves over number fields can be reduced to the case of elliptic curves. In this case we prove that, as a consequence of a result of Stoll, a section of the exact sequence of the absolute Galois group of an elliptic curve over a number field arises from a rational point if and only if the induced section of the corresponding (geometrically abelianised) arithmetic fundamental group of the elliptic curve arises from a rational point. We also prove that given any curve over a number field, there exists a double covering of this curve for which the birational setion conjecture holds true.