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### When do tamely ramified Belyi maps exist in characteristic p?

Let $p$ be a prime. Let $d$ be a number. Let $\lambda_0,\lambda_1,\lambda_{\infty}$ be three partitions of $d$ whose parts are prime to $p$. Consider the following question:
Does there exist a ...

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### Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...

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### technical question about a paper on Belyi's theorem

I'm trying to understand Belyi's theorem, as presented here: http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf
He defines a curve $X$ over a field $C$ to be a smooth projective geometrically connected ...

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### Curves which are not covers of P^1 with four branch points

The following interesting question came up in a discussion I was having with Alex Wright.
Suppose given a branched cover C -> P^1 with four branch points. It's not hard to see that the field of ...