4
$\begingroup$

A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. For any positive integer $n$, let $H_n^o$ be the moduli space of covers $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ of degree $n$ which are etale over $\mathbf{P}^1_{\overline{\mathbf Q}}-\{0,1,\infty,\lambda\}$, where $\lambda$ is allowed to vary in $\overline{\mathbf Q}-\{0,1\}$.

Q1. How many connected components does $H_n^o$ have? (I'm looking for a formula in terms of $n$.)

There is a finite etale morphism $\pi_n^0:H_n^o \to \mathbf{P}^1_{\overline{\mathbf{Q}}} \backslash \{0,1,\infty\}$ which sends the class of a cover $f:X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ to the unique element $\lambda$ in its branch locus different from $0$, $1$ and $\infty$.

Q2. The degree of $\pi_n^o$ is the number of subgroups of the fundamental group of $\mathbf{P}^1(\mathbf{C})-\{0,1,\infty,\lambda\}$ of index $n$. Is there a nice formula for this?

$\endgroup$

2 Answers 2

3
$\begingroup$

There is not going to be a nice answer to your Q1, because the decomposition of that Hurwitz space into components can be quite wild. In fact, any algebraic curve over Qbar is birational to a connnected component of one of these spaces! (See Diaz-Donagi-Harbater, "Every curve is a Hurwitz space.")

As for your Q2, these numbers are called "Hurwitz numbers" and a lot is known about them. Often one splits things up by specifying the conjugacy class of monodromy around the branch points. But given that you are NOT doing that, I think you are essentially just asking for the number of triples of elements of S_n which generate a transitive permutation group. I imagine you could do that by some kind of inclusion-exclusion argument. Or look up "subgroup zeta function" for the very rich general story about counting subgroups of index n in finitely generated groups.

$\endgroup$
2
$\begingroup$

Question 2 is answered in more detail than you could possibly want in Lubotzky's paper. (since your fundamental group is just a free group on three generators)

$\endgroup$

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.