The paper formulates a precise relationship between the Tate–Shafarevich group of an elliptic
curve
over
with a quotient of
the class group of
on which
operates by its standard 2-dimensional representation over
. We
establish such a relationship in many cases.
Keywords
class group, Tate–Shafarevich group, Selmer group, elliptic
curve, Galois cohomology