I had probed friends of mine about Grothendieck's motivation for making the anabelian geometry conjectures, and they gave me the following explanation:
If $X$ is a hyperbolic curve over some field $K$ (think projective and of genus $\geq 2$), then, intuitively, its universal cover is the upper half plane. This means that to distinguish between any two hyperbolic curves, it suffices to distinguish between the actions on the upper-half plane that induce those two hyperbolic curves. In some vague way, this should be the same as distinguishing between their fundamental groups.
This seems a little tenuous to me. Is there a modification of the above argument that gives a moral reason for why anabelian geometry should be correct? Is there a completely different moral reason for anabelian geometry? If so, what is it? What intuitive reason should I have to believe anabelian geometry (beside the mounting evidence that it is indeed true)?
Best Answer
I can only offer a "strengthening" of your friends' explanation. Let me first remark that I am not an expert in this field and I am sure that there are some grave mistakes in my argument. However, it is much too long for a comment, so I post it as an answer.
Let us first consider the simpler case of (co)homology instead of fundamental groups. When talking about étale (say, $\ell$-adic) cohomology together with its Galois action, the transcendental analogue is generally taken to be not just the singular cohomology groups, but these groups together with their Hodge structure.
Similarly, consider a hyperbolic curve $X$ over a number field $K$. For simplicity, assume we are given a $K$-rational base point $x\in X(K)$. The fundamental group one considers is either the group $\pi_1^{et}(X,x)$ as an abstract profinite group, or the group $\pi_1^{et}(X\otimes\overline{K},x)$ together with its action of $\operatorname{Gal}(\overline{K}|K)$. The former can be reconstructed from the latter. The weakest version of Grothendieck's anabelian conjecture for curves says (roughly) that we can reconstruct $X$ from $\pi_1^{et}(X,x)$.
Let me explain why we can reconstruct $X$ from $\pi_1^{et}(X\otimes\overline{K},x)$ with its Galois action. The abelianization of this group with Galois action is just the product over all $\ell$ of the $\ell$-adic Tate modules $T_{\ell }(\operatorname{Jac}X)$. These are the $\ell$-adic analogues of the Hodge structures on first homomology, which bear the same information as the Jacobian itself. Thus it is not surprising (although very difficult!) that we can reconstruct $\operatorname{Jac}X$ from these data, and the Jacobian determines the isomorphism class of the curve by Torelli's theorem. [Edit: As Torsten Ekedahl has pointed out in the comments, it is not true that you can recover an abelian variety from its Tate module.]
Now there are certainly some points where the above argument does not work as simply as presented, but the morals is that the analogue of the arithmetic fundamental group over $\mathbb{C}$ should be the topological fundamental group with a "Hodge structure on groups". I do not know if this has ever been worked out, but there is a good understanding of the "Hodge structure on the nilpotent completion of the fundamental group", introduced by Hain and Zucker.