# Constructing ramified covers with prescribed multiplicities at ramification points

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?

• The Riemann-Hurwitz formula is a numerical constraint on existence of such a cover. By the Riemann-Hurwitz formula, for a finite cover, the sum over all points $P$ of $m_P-1$ must be even. Thus, for instance (and this is obvious topologically), there is no cover of $\mathbb{P}^1$ that is branched simply over a single point. Jun 2, 2017 at 9:13
• I assume the characteristics is negative. I don't understand your argument on the RH formula. The sum goes over $e_P$ when $P$ in the cover not in $Y$. Hence, one may have many points lying over the same poins and having the same ramification. Jun 2, 2017 at 9:49
• Over $\mathbb{C}$, this amounts to find a transitive representation of the fundamental group of the complement of the points into the simmetric group $\mathbb{S}_n$, for a suitable $n$. Jun 2, 2017 at 11:05
• I misunderstood your question. As Francesco Polizzi points out, you are asking for a transitive representation of the fundamental group of the punctured Riemann surface in the symmetric group $\mathfrak{S}_n$, $n$ divisible by every $m_P$, such that the homotopy class of a loop around $P$ maps to an element of cycle type $(m_P^{n/m_P})$ with the standard notation for cycle types. For instance, when $Y$ is $\mathbb{P}^1$ and there are three points with $m_P=4$, this seems difficult . . . Jun 2, 2017 at 12:11
• If you allow a "few" simple branch points with branching $(2,1^{n-2})$ in $Y$ in addition to the points $P$ with branching $(m_P^{n/m_P})$, existence follows from work of Deligne, cf. Kluitman's article in the Braids volume. Jun 2, 2017 at 12:15

Theorem A of the following article of Pete Clark and Jon Voight seems to prove the existence of $X\to Y$ when $Y$ is $\mathbb{P}^1$ with three special points with multiplicities $(a,b,c)$.