Mordell-Weil group ( $E (K)$ )

K

E (K)

is a finitely generated abelian group.

Kummer map of an elliptic curve

n

sequence gives an injective connecting map

E (K) / n E (K) ↪ H^{1} (K, E [n]) .

Local conditions restrict its image to the finite

n

When

μ_{n} \subset K

σ \in Gal (L / K)

and

[x] \in K^{*} / (K^{*})^{n}

σ (n x) / n x \in μ_{n}

. Its elliptic analogue sends

σ

and

[P] \in E (K) / n E (K)

σ (Q) - Q \in E [n]

for

n Q = P

K

S

of finite primes,

K (S, n) = {[x] \in K^{*} / (K^{*})^{n} : v_{p} (x) \equiv 0 (mod n) for every p \in / S} .

For

E : y^{2} = x^{3} + a x^{2} + b x

and its two-isogenous curve

E^{'} : y^{2} = x^{3} - 2 a x^{2} + (a^{2} - 4 b) x

E (Q)

and

E^{'} (Q)

determine the Mordell-Weil rank.

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