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 *η*^{2} = 4*ξ*^{3} − *g*_{2}*ξ* − *g*_{3} or for simplicity

η^{2} = ξ^{3} + aξ + b |
(A.1) |

with fixed *a*, *b* such that

4a^{3} + 27b^{2} ≠ 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 ...

Start Free Trial

No credit card required