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Let $a$ be a closed point in $\mathbf{P}^1_{\overline{\mathbf{Q}}}$.

Let $Y \cong \mathbf{P}^1_{\overline{\mathbf{Q}}} $ and let $\pi:Y\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ be a finite morphism which is unramified above $a$. Let $b$ be a point in $\pi^{-1}(a)$.

Can one effectively bound the height of $b$ in terms of the degree of $\pi$, some data depending on the branch points of $\pi$ and the height of $a$?

Ineffectively this should be possible.

Maybe it's more natural to ask if one can bound the height of $\pi^{-1}(a)$ in terms of the degree of $\pi$, some data depending on the branch points of $\pi$ and the height of $a$.

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  • $\begingroup$ What do you mean by the "naive height of b" ? (a Weil height depends on a line bundle). ALso, I think $\pi=f$ in your post. $\endgroup$
    – Damian Rössler
    Commented Oct 3, 2011 at 9:10
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    $\begingroup$ Since you assume $Y\simeq \mathbf P^1$, it is more natural to give yourself $\pi$ as a rational function of some degree $d$. Then, there are effective constants $c$ and $c'$ such that $ d h(y) -c \leq h(\pi(y)) \leq d h(y)+c'$ for all $y\in \mathbf P^1(\overline{\mathbf Q})$. The constant $c'$ is easy to obtain. The effectivity of $c$ is more subtle (one can get one through resultants, but in a more general setting, some form of effective Nullstellensatz is necessary). $\endgroup$
    – ACL
    Commented Oct 3, 2011 at 21:09

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Let me reformulate the problem in less fancy terms. Let $\pi(x)\in\overline{\mathbb{Q}}$ be a rational function of degree $d\ge1$. As ACL noted, there is a standard estimate $$ dh(y) - c_1(\pi) \le h(\pi(y)) \le dh(y) + c_2(\pi), \quad (*) $$ where it is relatively easy to give explicit formulas for $c_1$ and $c_2$ in terms of $d$ and the coefficients of the polynomials defining $\pi$. However, what you seem to be asking is whether we can give formulas for $c_1$ and $c_2$ that depend only on $d$ and the branch points of $\pi$.

I suspect that the answer is no, for the following reason. For simplicity, lets assume that $\pi(\infty)\ne0$, so I can normalize $\pi$ so that its numerator is monic of degree $d$. A rational function of degree $d$ has $2d-2$ branch points, but with this normalization, it has $2d+1$ free coefficients. So fixing the branch points should give a 3 dimensional family of maps with those branch points. By varying $\pi$ within such a family, the values of $c_1(\pi)$ and $c_2(\pi)$ that come from the usual proof will not be bounded. This makes me suspect that by varying $\pi$ within the family, the inequality $(*)$ will fail if we require $c_1$ and $c_2$ to be fixed.

This isn't a proof, of course, but I hope it suggests where one might look for a negative answer to your second question.

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