Covers of the projective line over Z and arithmetic Grauert-Remmert - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:14:30Z http://mathoverflow.net/feeds/question/23724 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23724/covers-of-the-projective-line-over-z-and-arithmetic-grauert-remmert Covers of the projective line over Z and arithmetic Grauert-Remmert Ariyan Javanpeykar 2010-05-06T14:38:51Z 2011-01-30T09:19:15Z <p>This question is the two-dimensional analogue of <a href="http://mathoverflow.net/questions/22883/etale-coverings-of-certain-open-subschemes-in-spec-o-k" rel="nofollow">http://mathoverflow.net/questions/22883/etale-coverings-of-certain-open-subschemes-in-spec-o-k</a></p> <p>There I asked if one could characterize in a way certain covers of $\textrm{Spec} \ O_K$. As Cam Mcleman answered, this is basically done by the Galois group of the maximal extension unramified outside $D$. A covering of $U$ is of the form $O_L[\frac{1}{D}]$, where $L$ is any extension of $K$. </p> <p>Here I would like to ask the same question, only now for $X=\mathbf{P}^1_{\mathbf{Z}}$. </p> <p>Let $D$ be a normal crossings divisor on $\mathbf{P}^1_{\mathbf{Z}}$ and let $U$ be the complement of its support.</p> <p><strong>Q1</strong>. Is there an "equivalence of categories" as Georges Elencwajg mentions in his answer for the analytic case. (See above link.) Basically, is there an arithmetic Grauert-Remmert theorem? </p> <p><strong>Q2</strong>. What is known about the etale fundamental group in this case? Is it "finitely generated"? Has anybody studied the maximal pro-p-quotients of these groups? </p> <p><strong>Q3</strong>. The analytic analogue would be to consider the same question for $\mathbf{P}^1_{\mathbf{C}} \times \mathbf{P}^1_{\mathbf{C}}$. </p> <p><strong>Q4</strong> Lars (see above link) mentions a result for <code>$\mathbf{P}_{\mathbf{Q}}^1$</code>. Is there something similar for $\mathbf{P}^2_{\mathbf{Q}}$?</p> http://mathoverflow.net/questions/23724/covers-of-the-projective-line-over-z-and-arithmetic-grauert-remmert/30347#30347 Answer by Lars for Covers of the projective line over Z and arithmetic Grauert-Remmert Lars 2010-07-02T20:17:27Z 2010-07-02T20:17:27Z <p>Regarding Q3: For any scheme $X$ of finite type over $\mathbb{C}$ the Riemann-Existence Theorem (See SGA1 XII.5) says that the category of finite étale coverings of $X$ is equivalent to the category of finite covering spaces of the associated analytic space $X^{an}$. This implies that the finite quotients of the topological fundametal group of $X^{an}$ are the same as the finite quotients of the étale fundamental group, and one obtains that the étale fundamental group of $X$ is isomorphic to the profinite completion of the topological fundamental group of $X^{an}$.</p> <p>Q4: The same short exact sequence that I mentioned in your other question is still valid. </p> <p>As in your other question, I cannot say anything about the situation over $\mathbb{Z}$ :)</p> http://mathoverflow.net/questions/23724/covers-of-the-projective-line-over-z-and-arithmetic-grauert-remmert/53752#53752 Answer by Emerton for Covers of the projective line over Z and arithmetic Grauert-Remmert Emerton 2011-01-30T01:18:55Z 2011-01-30T01:18:55Z <p>This is a question which is too long to put in a comment box: What exactly do you mean by a simple normal crossings divisor in $\mathbb P^1_{\mathbb Z}$? </p> <p>Let me recall that an irreducible divisor in $\mathbb P^1_{\mathbb Z}$ consists either of a Galois conjugacy class of $\overline{\mathbb Q}$ points of $\mathbb P^1$ (closed up to make a divisor in $\mathbb P^1_{\mathbb Z}$ (these are the horizontal divisors), or else one of the closed fibres $\mathbb P^1_{\mathbb F_p}$ of the map $\mathbb P^1_{\mathbb Z} \to \mathbb Z$ (these are the vertical divosors). </p> <p>The vertical divisors are mutually disjoint, while the horizontal divisors meet each vertical divisor in a point (if the horizontal divisor is the point $\alpha \in \overline{\mathbb Q} \cup {\infty}$, and the vertical divisor is $\mathbb P^1_{\mathbb F_p}$, then they intersect in the point $\overline{\alpha}$ obtained by specializing $\alpha$ into char. $p$). Two horizontal divisors, say corresponding to the points $\alpha, \beta \in \overline{\mathbb Q}\cup \infty$, may or may not intersect; they meet in a point lying over the point $p \in$ Spec $\mathbb Z$ if and only if $\alpha$ and $\beta$ have the same specialization to char. $p$, and one can define a multiplicity of intersection, which reflects the precise power of $p$ modulo which they are congruent.</p> <p>I'm not exactly sure in this arithmetic context what the s.n.c. condition is.</p>