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

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 $\Bbb Q$ 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.Comment: Final versio

On Building superpotentials in F- GUTs

Using characters of finite group representations, we construct the fusion algebras of operators of the spectrum of F- theory GUTs. These fusion relations are used in building monodromy invariant superpotentials of the low energy effective 4d $\mathcal{N}=1$ supersymmetric GUT models.Comment: LaTeX, 32 pages, 1 figur