This is motivated by a basic number theory question I asked the previous day: Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$? I noted there that the answer to the corresponding question is "no" in complex function fields, i.e. for Riemann surfaces, and the reason is just that we are able to draw paths in the Riemann sphere, having as consequence that the fundamental group of $\widehat{\mathbb{C}} - S$ depends only on the number of punctures $|S|$.
The vague part of this question is this: what meaning, if any, has or can be ascribed to the idea of a path in $\mathrm{Spec} \, \mathbb{Z}$? Of course the groups $G_p := $ Galois group of the maximal unramified outside $p$ extension of $\mathbb{Q}$, are not all the same: for some $p$ there are cubic fields unramified outside $p$, for others, there aren't. But are they close to one another as abstract groups, as the idea of "connecting two primes by a path" would suggest? For instance, is it thinkable that $c_3(d) := \mathrm{dim}_{\mathbb{F}_3} \mathrm{Cl}(\sqrt{d}) \otimes \mathbb{F}_3$ might be bounded? (Recall by class field theory that $c_3(d)$ controlls the number of cubic fields with a given discriminant $d$; then a cubic field unramified outside $p$ has only a handful of possibilities $\pm p^{a}, \, a \leq 6$, for its discriminant. )
[Answer: No, this contradicts the Cohen-Lenstra heuristics, as Michael Stoll notes in the comments below. The $c_3(d)$ are bounded on average, by Davenport and Heilbronn's classic result, but it is expected following Cohen and Lenstra that for every $r$ the set of $d$ with $c_3(d) = r$ should have a certain definite density $> 0$. So the number of degree three covers of a punctured $\mathrm{Spec} \, \mathbb{Z}$ cannot be bounded by the number of punctures alone; we need also some input from the sizes of the punctures. On the other hand $3^{c_3(d)} \ll_{\varepsilon} d^{\varepsilon}$ is a well known hypothesis, with current record of $\varepsilon = 0.223\ldots$ due to Helfgott and Venkatesh as far as I am aware. Perhaps generalizing this conjecture there could be a sense in which $G_p$ are "close" as abstract groups, in some quantitative sense involving an $\varepsilon$, viz. $p^{\varepsilon}$... ]
Anyway I had the following completely specfic question, looking to interpret the observation that various results in diophantine approximations (such as the number of solutions to $S$-unit equations, exceptions to Roth's inequality, etc.) show a dependence on $S$ only through the number of punctures $|S|$. It is about Kim's approach to the $S$-unit equation: Does it match the diophantine results on the number of solutions? Here is what I wrote initially:
While I am not at all confident that any sense can go into the idea of paths in $\mathrm{Spec} \, \mathbb{Z}$, there is the related idea of the motivic fundamental group, which is perfectly on solid ground for a unirational variety over $\mathbb{Q}$. For $\mathbb{P}^1 - \{0,1,\infty\}$ the point of view of the motivic fundamental group has been used by Minhyong Kim to give an apparently completely new proof of the finiteness of solutions in $S$-units to the equation $x+y = 1$. All the other proofs are essentially diophantine. If I understand correctly Kim's approach is specific to the case of $\mathbb{Q}$, but still I wondered if it could give any insight on the above kind of questions.
Hence I wanted to ask concretely the following. If none of the rest makes any sense, please take this as the real
Question. Does Kim's proof of the finiteness of solutions to the $S$-unit equation produce a bound exponential in $|S|$? (Indeed, is this proof even effective?)
I would like to study Kim's proof, but I am only beginning to do so; hence, at the moment I have no clue what the answer to this question is. The diophantine approximations approach does produce exactly such a bound, as would be expected by the idea of paths.
