422 research outputs found
On the section conjecture over function fields and finitely generated fields
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 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.Comment: Final versio
Arithmetic of -adic curves and sections of geometrically abelian fundamental groups
Let be a proper, smooth, and geometrically connected curve of genus
over a -adic local field. We prove that there exists an
effectively computable open affine subscheme with the property
that , and equals or (resp. , assuming ), if (resp. if and only if) the exact
sequence of the geometrically abelian fundamental group of splits. We
compute the torsor of splittings of the exact sequence of the geometrically
abelian absolute Galois group associated to , and give a new
characterisation of sections of arithmetic fundamental groups of curves over
-adic local fields which are orthogonal to (resp. ).
As a consequence we observe that the non-geometric (geometrically pro-)
section constructed by Hoshi in [Hoshi] is orthogonal to .Comment: To appear in Mathematische Zeitschrif
A local-global principle for torsors under geometric prosolvable fundamental groups
We prove a local-global principle for torsors under the prosolvable geometric
fundamental group of a hyperbolic curve over a number field
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