On the section conjecture over function fields and finitely generated fields
Publications of the Research Institute for Mathematical Sciences
European Mathematical Society
This is the author accepted manuscript. The final version is available from the European Mathematical Society via the DOI in this record.
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 Q if it holds over all number fields, under the condition of finiteness (of the -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.
Vol. 52, No. 3, pp. 335–357