what does it mean for a rational number to be "complicated"?

Question

I was reading about the proof of the Mordell-Weil Theorem in "Rational points on elliptic curves" from Silverman and Tate and in Chapter III is the hight function described as an indicator on how "complicated" a rational number is. Does this mean that if the hight function of a rational number has greater value that it requires more numbers to write it?

Answer 1

Roughly speaking, we can measure how "complicated" an algebraic object is by the number of bits it takes to describe the object. So in that sense, a rational number $a/b$ takes $\log_2|a|$ bits to describe the numerator and $\log_2|b|$ b its to describe the denominator, plus another bit to specify the sign. So a reasonable measure of the complexity of a rational number would be $$ \text{Complexity}(a/b) = \log_2|a| + \log_2|b| + 1. \tag{$*$}$$ (For $a=0$, I'll just set the complexity to be $1$.) Anyway, the first crucial property we want a complexity measure to have is that there are only finitely many objects of complexity less than any specified bound. A second useful property is that we have some understanding of how complexity changes when we apply a "nice" function. Having said that, if $h$ is a good complexity measure, than for example, $2h+1$ is also likely to work fine; so there's some flexibility. Then, sometimes, one can find a particular complexity measure that transforms very nicely for particular kinds of maps, and using those leads to nice formulas (and other treats). In particular, the complexity measure $(*)$ for rational numbers is okay, but $$ h(a/b) = \log\max\bigl\{ |a|,\,|b| \bigr\} $$ has the nice property that $$ h(\alpha^d) = |d|\cdot h(\alpha)\quad\text{for all $\alpha\in\mathbb Q^*$ and all $d\in\mathbb Z$.}$$ And we have inequalities $$ \frac1{10}h(\alpha)-1 \le \text{Complexity}(\alpha) \le 10h(\alpha)+1, $$ so in terms of the finiteness property, they're both more-or-less the same. (I used $10$ and $\frac{1}{10}$ because I'm lazy; you can certainly find better constants!)