There are other abelian groups attached to elliptic curves, such as
the torsion subgroup
of elements of of finite
order. The torsion subgroup is (isomorphic to) the group that
appeared in Equation (10.1.2) above). When is a number
field, there is a group called the Shafarevich-Tate group
attached to , which plays a role similar to that of the class group
of a number field (though it is an open problem to prove that
is finite in general). The definition of
involves Galois
cohomology, so we wait until Chapter 11 to define it. There
are also component groups attached to , one for each prime of
. These groups all come together in the Birch and
Swinnerton-Dyer conjecture (see `http://wstein.org/books/bsd/`).

William Stein
2012-09-24