On the Galois cohomology of elliptic curves defined over a local field

In this article we calculate the group of principal homogeneous spaces over elliptic curves defined over a complete discretely normed field with algebraically closed residue field of characteristic $p>3$ and belonging to types $(c\,4)$, $(c\,5)$ in the classification of A. Neron. The result of our calculations refutes Neron's earlier statement that the group of principal homogeneous spaces over curves of type $(c)$ is trivial. Moreover the calculation of the fundamental group of the proalgebraic group of the points on these curves that are rational over the ground field supports (in this case) the conjecture of I. R. Shafarevich concerning the duality of the group of principal homogeneous spaces and the character group of the fundamental group.
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