On the section conjecture over function fields and finitely generated fields

TL;DR

This paper explores the section conjecture for hyperbolic curves over function and finitely generated fields, establishing conditions under which the conjecture over number fields implies its validity over broader classes of fields.

Contribution

It proves that the anabelian section conjecture over finitely generated fields follows from its validity over number fields, assuming finiteness conditions on Shafarevich-Tate groups.

Findings

Section conjecture holds over all finitely generated fields if it holds over all number fields under certain conditions.

The conjecture's validity over number fields implies its validity over all finitely generated fields for curves defined over number fields.

Abstract

We investigate sections of arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over $\Bbb Q$ if it holds over all number fields, under the condition of finiteness (of the $\ell$-primary parts) of certain Shafarevich-Tate groups. We also prove that if the section conjecture holds over all number fields then it holds over all finitely generated fields for curves which are defined over a number field.