3
$\begingroup$

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?

$\endgroup$
6
  • 2
    $\begingroup$ 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. $\endgroup$ Jun 2, 2017 at 9:13
  • 2
    $\begingroup$ 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. $\endgroup$
    – ralleee
    Jun 2, 2017 at 9:49
  • 1
    $\begingroup$ 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 $n$. $\endgroup$ Jun 2, 2017 at 11:05
  • 1
    $\begingroup$ 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 . . . $\endgroup$ Jun 2, 2017 at 12:11
  • $\begingroup$ 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. $\endgroup$ Jun 2, 2017 at 12:15

1 Answer 1

3
$\begingroup$

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 $(a,b,c)$.

Pete L. Clark and John Voight
Algebraic Curves Uniformized by Congruence Subgroups of Triangle Groups
https://arxiv.org/pdf/1506.01371.pdf

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.