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. When $K$ is a number field, there is a mysterious group called the Shafarevich-Tate group ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)$ attached to $E$. It's definition 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 Conjecture ).