The Shafarevich-Tate Group and the Mordell-Weil Theorem

Definition 2.8 (Shafarevich-Tate Group)   The Shafarevich-Tate group of an elliptic curve $E$ over a number field $K$ is

$${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\... ...}}}(E/K) = \ker\left(\H^1(K,E) \to \prod_v \H^1(K_v,E)\right).$$

For any positive integer $n$, we may thus add in a row to (2.2.2):

$$\xymatrix @=1.2em{ 0 \ar[r]& {E(K)/n E(K)} \ar[r]\ar[d]& {\... ... \H^1(K_v,E[n])} \ar[r]& {\prod_v H^1(K_v,E)[n]} \ar[r]& 0. }$$

The $n$-descent sequence for $E$ is the short exact sequence

$$0 \to E(K)/n E(K) \to \Sel ^{(n)}(E/K) \to {\mbox{{\fontenc... ...yr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/K)[n] \to 0.$$ (2.2.3)

Theorem 2.9   For every integer $n$ the group $\Sel ^{(n)}(E/K)$ is finite.

Exercise 2.10   Prove the that $E[n]$ is a finite Galois extension of $K$.

Theorem 2.11 (Mordell-Weil)   The group $E(\mathbb{Q})$ is finitely generated.