On elliptic curves over pseudolocal fields

In this paper it is shown that for pseudolocal fields there is a natural analog of the Tate–Shafarevich duality for elliptic curves, taking the following form:
Theorem. If $A$ is an elliptic curve defined over the pseudolocal field $k$, whose residue field has characteristic not equal to $2$ or $3$, then the Tate–Shafarevich pairing
$$ H^1(k,A)\times A_k\to Q/Z $$
is left nondegenerate.
Bibliography: 11 titles.