Other Groups

There are other abelian groups attached to elliptic curves, such as the torsion subgroup $E(K)_{\tor }$ of elements of $E(K)$ of finite order. The torsion subgroup is (isomorphic to) the group $T$ that appeared in Equation (10.1.2) above). When $K$ is a number field, there is a group called the Shafarevich-Tate group ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$ attached to $E$, which plays a role similar to that of the class group of a number field (though it is an open problem to prove that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$ is finite in general). The definition of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$ involves Galois cohomology, so we wait until Chapter 11 to define it. There are also component groups attached to $E$, one for each prime of $\O_K$. These groups all come together in the Birch and Swinnerton-Dyer conjecture (see http://wstein.org/books/bsd/).