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 believe anabelian geometry (beside the mounting evidence that it is indeed true)?

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)?