Let $Y$ be a smooth projective curve defined over number field $K$. Let $P_1,\dots ,P_m$ be some $K$-points to which we will associate "multiplicities" $m_i\in\{ 2,3,\dots \}$ for $i=1,\dots ,m$. The other $K$ points (and even those not defined over $K$ ) are considered with "multiplicities" $1$. We want to find a smooth projective curve $X$ defined over some number field $M$, and a morphism $\phi :X\to Y$ such that $e_{T}=m_{\phi (T)}$ for all closed points $T$ of $X$. Here $e_T$ is the ramification at the point $T$. We assume the "characteristics" of $Y$ is negative, that is $2-2g-\sum _P(1-\frac{1}{m_P})<0$. Here $g$ is the genus.

In Serre's Topics in Galois theory, he solves the problem for the Riemann surfaces. He doesn't require that the first curve is projective. The universal cover is $\mathbb{H}$, the Poincaré half-plane, if the "characteristics" is negative.

The cover here is infinite. Of course, I need a finite. Therefore somehow I should quotient it by some group and get a smooth projective curve. Then I know how to use GAGA to finish the problem.

Any ideas how to make the missing step?

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