In short, unlike in the case $n = 2$1$, we can't hope to find groups that will be related in some direct manner (unlike in the case$n = 1$, where class field theory gives a direct relation between$G_K^{ab}$and$K^{\times}\backslash \mathbb A^{\times}$); the relation(in fact, impossible) to separate the archimedean and non-archimedean primes. If you play around and follow up some of the references suggested by others, you will see (in the case$K = \mathbb Q$) that this quotient is related to the upper half-plane and congruence subgroups, and the generating vectors for automorphic representations are precisely primitive Hecke eigenforms (either holomorphic modular forms or else Maass forms). The archimedean primes contribute prime contributes the upper half-plane, and the finite primes contribute the level of the congruence subgroup and the Hecke operators. give compatible systems of two-dimensional$\ell$-adic representations. The general statement here is due to Deligne, and can be learnt (at least as at first) as a black-box. The case of weight 2, which includes the case of elliptic curves over$\mathbb Q$, is easier, and it's possible to learn the whole story (other than proof of the modularity theorem itself) in a reasonable amount of time. The case of weight one is special, and is treated in a beautiful paper of Deligne and Serre. The converse here (that every two-dimensional 3 added 53 characters in body The Galois side involves studying continuous$n$-dimensional representations of the Galois group ofhence what are usually called Artin representationrepresentations. on the quotient$GL_n(K) \backslash GL_n(\mathbb A)$. (To provide some orientation with regardd regard to this notion: Note that, by Frobenius reciprocity,of the representation) we get a Hecke eigenvalue. The matching is given by the rule that traces of Frobenius elements should equal Hecke eigenvalues. theory for locally compact abelian groups, and speaking somewhat loosely) the sum of spaces spanned by characters, hence so that automorphic representations are just characters of(But it is better to regard these the latter as corresponding more generally corresponding to idele class characters --- also called Hecke characters or Grossencharacters; these are not-necessarily-finite-order generalizations of Dirichlet characters.) As the preceding summary shows, the big picture here is pretty big, and the technical details are quite extensive and involved. There is the added complication that the proofs of what is known in the non-abelian case are very involved, and often use methods that don't play a role in the general story, but seem to be crucial for the arguments to go through. (E.g. Mellin transforms don't play any role in the general theory of attaching$L$-functions to automorphic representations. But , but in the case$n = 2$, the fact that you can pass from a modular form to its$L$-function via a Mellin transform is often useful and important.) but which are currently the only known methods for making progress (depending on what direction you want to pursue: , these include the trace formula, Shimura varieties, the deformation theory of Galois representations, the Taylor--Wiles method, ... ). 2 added 10 characters in body The quotient$GL_2(K)\GL_2(\mathbb GL_2(K)\backslash GL_2(\mathbb A)\$ is not a group,