2 (Edited to fix formatting. Thanks, Zev.)

Hi Kevin,

I proved that if E/Q $E/\mathbf{Q}$ is given using by a minimal Weiestrass equation, then

$#E(Z) \#E(Z) \le C^{rank C^{\text{rank} E(Q) + n(j) + 1}$

where n(j) $n(j)$ is the number of distinct primes dividing the denominator of the j-invariant $j$-invariant of E $E$ and C $C$ is an absolute constant. This is in J. Reine Angew. Math. 378 (1987), 60-100.

Mark Hindry and I proved that if you assume the abc conjecture, then you can remove the n(j) in the above estimate. This is in Invent.~MathInvent. Math. 93 (1988), 419-450. It is a conjecture due to Lang.

The papers contain more general results for (quasi)-S-integral points over number fields.

1

Hi Kevin,

I proved that if E/Q is given using by a minimal Weiestrass equation, then

$#E(Z) \le C^{rank E(Q) + n(j) + 1}$

where n(j) is the number of distinct primes dividing the denominator of the j-invariant of E and C is an absolute constant. This is in J. Reine Angew. Math. 378 (1987), 60-100.

Mark Hindry and I proved that if you assume the abc conjecture, then you can remove the n(j) in the above estimate. This is in Invent.~Math. 93 (1988), 419-450. It is a conjecture due to Lang.

The papers contain more general results for (quasi)-S-integral points over number fields.