On the Section Conjecture over Function Fields and Finitely Generated Fields

Abstract

We investigate sections of the 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 $\mathbb{Q}$ if it holds over all number fields, under the condition of finiteness (of the $l$-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.