12   Elliptic Curves over Number Fields  The weak Mordell Weil Theorem  Theorem 12 1  Assuming that   E   K an elliptic curve  L   K a finite Galois extension  Then  the natural map E   K   n E   K     E   L   n E   L   has finite kernel   Proof  For each element in the kernel we pick a coset representative P   E   K   and then Q   E   L   such that n Q   P   For any     Gal   L   K   we have      n     Q   Q       P   P   0       So    Q   Q   E   n     Since Gal   L   K   and  E   n   are finite  there are only finitely many possibilities for the map     Gal   L   K     E   n         Q   Q      even without requiring it to be a group homomorphism     So we have a map     ker   E   K   n E   K     E   L   n E   L             Maps   Gal   L   L     E   n     P   n E   K             Q   Q   where n Q   P     It remains to show     is injective   So suppose P 1   P 2   E   K    P i   n Q i for i   1   2  and suppose   Q 1   Q 1     Q 2   Q 2 for all     Gal   L   K    Then     Q 1   Q 2     Q 1   Q 2 for all     Gal   L   K    hence Q 1   Q 2   E   K    so P 1   P 2   n E   K    Hence P 1   n E   K     P 2   n E   K   as desired     Theorem   Weak Mordell Weil Theorem   Assuming that   K a number field  E   K a elliptic curve  n   2 an integer  Then  E   K   n E   K   is finite   Proof  Theorem 12 1 tells us that we may replace K by a finite Galois extension   So without loss of generality   n   K and  E   n     E   K    Let      S         n       primes of bad reduction for E   K         For each P   E   K    the extension K     n     1 P     K is unramified outisde S by Theorem 9 8  Since Gal   K     K   acts on   n     1 P  it follows that Gal   K     K     n     1 P     is a normal subgroup of Gal   K     K   and hence K     n     1 P     K is a Galois extension   Let Q     n     1 P  Since  E   n     E   K    we have K   Q     K     n     1 P    Consider     Gal   K   Q     K     E   n           n     2       Q   Q     Group homomorphism       Q   Q         Q   Q         Q   Q     Injective   If   Q   Q then   fixes K   Q   pointwise  i e      1   Therefore K   Q     K is an abelian extension of exponent n  unramified outside S   Proposition 11 3 shows that as we vary P   E   K   there are only finitely many possibilities for K   Q     Let L be the composite of all such extensions of K  Then L   K is finite and Galois  and E   K   n E   K     E   L   n E   L   is the zero map  Theorem 12 1 gives   E   K   n E   K             Remark   If K     or    or   K     p       then   E   K   n E   K          yet E   K   is uncountable  so not finitely generated    Fact   If K is a number field  then there exists a quadratic form    canonical height       E   K         0 with the property that for any B   0           P   E   K         P     B           is finite   Theorem   Mordell Weil Theorem   Assuming that   K a number field  E   K an elliptic curve  Then  E   K   is a finitely generated abelian group   Proof  Fix an integer n   2   Weak Mordell Weil Theorem implies that   E   K   n E   K        Pick coset representatives P 1   P 2       P m  Let               P   E   K       P       max 1   i   m     P i           Claim     generates E   K     If not  then there exists P   E   K       subgroup generated by     of minimal height  exists by        Then P   P i   n Q for some i and Q   E   K    Note that Q   E   K       subgroup generated by       Minimal choice of P gives                4     P     4     Q     n 2     Q         n Q         P   P i         P   P i         P   P i     2     P     2     P i   parallelogram law     Therefore       P         P i    Hence P      by definition of     which contradicts the choice of P   This proves the claim   By         is finite