Let ξ be an algebraic number. In Proposition 14.10 for one variable we estimated the height of a rational function of ξ from above in terms of the height H(ξ) of ξ. In Exercise 14.77 we verified that the upper bound is usually asymptotically best possible as H(ξ) gets large. Here we find an analogous upper bound on a fixed elliptic curve; that is, we consider algebraic numbers ξ, η related by η2 = 4ξ3 − g2ξ − g3 or for simplicity
|η2 = ξ3 + aξ + b||(A.1)|
with fixed a, b such that
|4a3 + 27b2 ≠ 0.||(A.2)|
If a, b are also algebraic, then we want to estimate from above the height of a rational function in ξ and η in terms of H(ξ) or better its logarithm h(ξ).
This is in itself not difficult; for example ...