comment on why is it that we only consider curves and not higher dimensional varieties. The point is that while function fields of curves are very similar to number fields, the fields of functions on higher dimensional varieties have a very different structure. For example, if X is a smooth surface, then the completions of the field of rational functions on X are labeled by pairs: a point x of X and a germ of a curve passing through x. The corresponding complete field is isomorphic to the field of formal power series in two variables. At the moment no one knows how to formulate an analogue of the Langlands correspondence for the field of functions on an algebraic variety of dimension greater than one, and finding such a formulation is a very important open problem. There is an analogue of the abelian class field theory , i.e. higher class field theory, but not much is known beyond that.
An open question from V. Drinfeld.
Let $ X $ be an irreducible smooth variety over a finite field $ \mathbb{F}_{p} $ and let $ \mathcal{E} $ be an irreducible lisse $ \overline{\mathbb{Q}_{\ell}} $-sheaf with $ \ell\neq p $ of rank $ r $ on $ X $ whose determinant has finite order. Let $ K $ be the field of ration functions on $ X $ and let $ \rho $ be the $ \ell $-adic representation of $ \text{Gal}(\overline{K}/K) $ corresponding to $ \mathcal{E} $. Is it true that a certan Tate twist of $ \rho $ appears as a subquotient of $ H^{i}(Y\times_{K}\bar{K},\overline{\mathbb{Q}_{\ell}}) $ for some algebraic variety $ Y $ over $ K $ and some number $ i $? L. Lafforgue gave a positive answer to the above Question if $ \text{dim}(X)=1 $. Neverthless, it remains open if $ \text{dim}(X)>1 $.