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

Let $Y$ be a smooth projective curve over Y \cong \mathbf{P}^1_{\overline{\mathbf{Q}}} $ \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 $b$ (with respect to the canonical sheaf endowed with its Arakelov metric) 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. If we fix the degree and the branch locus, the number of such covers is finite.

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

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

Can one bound the (naive) height of $b$ (with respect to the canonical sheaf endowed with its Arakelov metric) 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. If we fix the degree and the branch locus, the number of such covers is finite.

Maybe it's more natural to ask if one can bound the naive 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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Comparing heights of rational points on curves through covers

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

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

Can one bound the (naive) 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. If we fix the degree and the branch locus, the number of such covers is finite.

Maybe it's more natural to ask if one can bound the naive 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 $a$.