I was reading Franz Lemmermeyer's introduction to Fermat's Last and Wiles' Theorem, where he states
Galois representations $\rho_p : G_\mathbb Q\rightarrow GL_2(\mathbb Z_p)$ are used for studying the hardly understood group $G_\mathbb Q$ via the images of homomorphisms in well understood groups like $GL_2(\mathbb Z_p)$. We can make the left hand side more manageable by cutting it into little $\ell$-adic pieces.
On the surface it makes alot of sense, as we read this regularly in such introductory texts or elsewhere. On the other hand we have very many conjectures on Galois representations and associated objects, which are widely believed to hold modulo small corrections: the Langlands correspondence and Langlands' principle of functoriality, the Fontaine-Mazur conjecture, the galoisian theory of motives included period conjectures, $L$-functions including equivariant aspects, some connections to 3-manifolds or even topological quantum field theory, some connections to dynamical systems, some to operator algebras, perhaps some aspects to computability or complexity theory via model theory,...
We also have the inverse Galois problem, where i don't know what people expect, but this part seems less predictable than the above (to me).
It would seem that all those tools offer avenues to extract most things of interest we can think of from Galois groups of arithmetic-geometric fields -at least for not-very-specific ones. But i do not see how proofs of those widely-believed conjectures would provide much more insight than the conjectures themselves -provide conjecturally, but as humans we are accustomed to using machinery/conjectures we have no idea how to prove.
Now, it makes sense that inverse Galois theory would tell us more about the nature of Galois groups and fields, so is this mostly where their mysteries still remain ? However it seems to elicit less interest (at least less publishing) than conjectures in arithmetic geometry, so perhaps Galois groups per se are not as interesting as their representation theory and derived quantities, are they ?
In terms of "work", i would expect many natural algorithms in Galois theory to be complex or universal -thus some questions about Galois groups undecidable. But how hard do we expect inverse Galois problems to be -whether we expect a "closed form" yes/no answer for a natural class of groups, or an algorithm given a presentation of a group to be realized as Galois ? -I don't ask for single conjectures like of Langlands type, that would deserve separate questions, but you may still answer on that.
Please share your comments and sentiment: how hopeful are you or others, how far (from ignorance) has the mathematical community moved towards a comfortable mastery of the subject, how much do you expect the basic philosophy of Galois groups to change in coming decades ? Thank you.
