  Assuming the finiteness of the Shafarevich-Tate group of elliptic curves over
number fields we make several observations on the birational Grotendieck
anabelian setion conjecture. We prove that the birational setion conjecture for
curves over number fields can be reduced to the case of elliptic curves. In
this case we prove that, as a consequence of a result of Stoll, a section of
the exact sequence of the absolute Galois group of an elliptic curve over a
number field arises from a rational point if and only if the induced section of
the corresponding (geometrically abelianised) arithmetic fundamental group of
the elliptic curve arises from a rational point. We also prove that given any
curve over a number field, there exists a double covering of this curve for
which the birational setion conjecture holds true.
