13   Heights  For simplicity  take K      Write P     n   Q   as P     a 0   a 1       a n   where a 0       a n     and gcd   a 0   a n     1   Definition   Height of a point   H   P     max 0   i   n   a i     Lemma 13 1  Assuming that   f 1   f 2       X 1   X 2   coprime homogeneous polynomials of degree d  let     F     1     1   x 1   x 2       f 1   x 1   x 2     f 2   x 1   x 2         Then  there exist c 1   c 2   0 such that for all P     1            c 1 H   P   d   H   F   P       c 2 H   P   d       Proof  Without loss of generality f 1   f 2       X 1   X 2     Upper bound   Write P     a 1   a 2    a 1   a 2     coprime      H   F   P       max     f 1   a 1   a 2         f 2   a 1   a 2         c 2 max     a 1   d     a 2   d       where      c 2   max i   1   2  sum of absolute values of coefficients of f i         Therefore   H   F   P       c 2 H   P   d   Lower bound   We claim that there exists g i j       X 1   X 2   homogeneous of degree d   1 and         0 such that        j   1 2 g i j f j     X i 2 d   1   i   1   2           Indeed  running Euclid s algorithm on f 1   X   1   and f 2   X   1   gives r   s       X   of degree   d such that      r   X   f 1   X   1     s   X   f 2   X   1     1       Homogenising and clearing denominators gives      for i   2  Likewise for i   1   Write P     a 1   a 2   with a 1   a 2     coprime       gives        j   1 2 g i j   a 1   a 2   f j   a 1   a 2       a i 2 d   1   i   1   2       Therefore gcd   f 1   a 1   a 2     f 2   a 1   a 2     divides gcd     a 1 2 d   1     a 2 2 d   1                a i 2 d   1     max j   1   2   f j   a 1   a 2           H   F   P       j   1 2   g i j   a 1   a 2           i H   D   d   1     where        i     j   1 2  sum of absolute values of coefficients of g i j         Therefore          a i   2 d   1     H   F   P       i H   P   d   1     so      H   P   2 d   1   max     1     2   H   F   P     H   P   d   1     so      1 max     1     2       c 1 H   P   d   H   F   P             Notation  For x       H   X     H     x   1       max     u       v      where x   u v  u   v     coprime    Definition   Height of a point   Let E     be an elliptic curve  y 2   x 3   a x   b   Define the height     H   E             1 P     H   x   if P     x   y   1 if P   0 E     Alsdefine logarithmic height     h   E             0 P   log H   P       Lemma 13 2  Assuming that   E  E   are elliptic curves over        E   E   an isogeny defined over    Then  there exists c   0 such that        h       P         deg     h   P       c   P   E             Note   c depends on E  and E    but not  P   Proof  Recall  Lemma 5 4     deg     deg      d say   Lemma 13 1 tells us that there exists c 1   c 2   0 such that      c 1 H   P   d   H       P       c 2 H   P   d   P   E             Taking logs gives        h       P       d h   P       max   log c 2   log c 1       c       Example        2     E   E  Then there exists c   0 such that        h   2 P     4 h   P       c   P   E             Definition   Canonical height of a point   The canonical height  is             P     lim n     1 4 n h   2 n P         We check convergence   Let m   n  Then       1 4 m h   2 m P     1 4 n h   2 n P         r   n m   1   1 4 r   1 h   2 r   1 P     1 4 r h   2 r P         r   n m   1 1 4 r   1   h   2   2 r P       4 h   2 r P       c   r   n   1 4 r   1   c 4 n   1 1 1   1 4   c 3   4 n   0     as n      So the sequence is Cauchy        P   exists   Lemma 13 3      h   P         P     is bounded for P   E         Proof  Put n   0  in above calculation to get        1 4 m h   2 m P     h   P       c 3       Take limit m         Lemma 13 4  Assuming that   B   0  Then             P   E             P     B             Proof        P   bounded means we have a bound on  h   P    by Lemma 13 3    So only finitely many possibilities for x  Each x gives   2 choices for y     Lemma 13 5  Assuming that       E   E   an isogeny defined over    Then                 P       deg         P     P   E             Proof  By     66  there exists c   0 such that        h     P       deg     h   P       c   P   E             Replace P by 2 n P  divide by 4 n and take limit n         Remark    i  Case deg     1 shows that      unlike  h  is independent of the choice of Weierstrass equation    ii  Taking       n     E   E shows               n P     n 2     P     n       P   E             Lemma 13 6  Assuming that   E     an elliptic curve  Then  there exists c   0 such that for all P   Q   E       with P   Q   P   Q   P   Q   0  we have      H   P   Q   H   P   Q     c H   P   2 H   Q   2       Proof  Let E have Weierstrass equation y 2   x 3   a x   b  a   b      Let P   Q   P   Q   P   Q have x coordinates x 1       x 4  By Lemma 5 8  there exists W 0   W 1   W 2       x 1   x 2   of degree   2 in x 2 such that        1   x 3   x 4   x 3 x 4       W 0       x 1   x 2   2   W 1   W 2         Write x i   r i s i with r i   s i      coprime        s 3 s 4   r 3 s 4   r 4 s 3   r 3 r 4     gcd   1       r 1 s 2   r 2 s 1   2       H   P   Q   H   P   Q     max     r 3       s 3     max     r 4       s 4       2 max     s 3 s 4       r 3 s 4   r 4 s 3       r 3 r 4       2 max     r 1 s 2   r 2 s 1   2         c max     r 1   2     s 1     2 max     r 2       s 2     2   c H   P   2 H   Q   2     where c depends on E  but not on P and Q     Theorem 13 7        E             0 is a quadratic form   Proof  Lemma 13 6 and     h   2 P     4 h   P     bounded gives that there exists c     such that      h   P   Q     h   P   Q     2 h   P     2 h   Q     c   P   Q   E             Replacing P  Q by 2 n P  2 n Q  dividing by 4 n and taking the limit n     gives                   P   Q         P   Q     2     P     2     Q         Replacing P   Q by P   Q  P   Q and         2 P     4     P   gives the reverse inequality  Therefore     satisfies the parallelogram law  and hence     is a quadratic form     Remark   For K a number field and P     a 0   a 1       a n       n   K    define      H   P       v max 0   i   n   a i   v     where the product is over all places v  and the absolute values are normalised such that        v       v   1       K         Using this definition  all results in this section generalise when   is replaced by a number field K