The (conjectural) Langlands correspondence for number fields gave rise to a Langlands correspondence for function fields (proved by Drinfeld and Lafforgue), where the most important difference is the absence of the infinite place which simplifies things in the latter. This, in turn, can be translated to a "geometric" Langlands correspondence for curves over fields, but there are certain differences.

First, I'm going to explain what the Langlands correspondence for function fields says, with the assumption in mind that you have been exposed to some algebra before. After that, I'm going to sketch how to go to the geometric Langlands correspondence, following Frenkel's storytelling in chapter 3 of his article "Langlands and conformal field theory". Of course, this being a blogpost, I won't repeat what Frenkel says (nor delve deeper) but try to summarise, to give an overview.

This post grew out of the preparation for a seminar talk on this topic and is separated in two parts, this being the first part. The continuation is here, discussing the geometrization.

Langlands Correspondence for Function Fields

I will now give you the correspondence and then spend a lot of time to explain all parts.

Let $X$ be a smooth projective curve, defined over $\mathbb{F}_q$, and $F = \mathbb{F}_q(X)$ its function field. Then, roughly, certain l-adic Galois representations correspond to certain automorphic representations. Less roughly, and more mathematically, this means:

There is a 1:1-correspondence between the irreducible continuous l-adic n-dimensional Galois representations that are unramified except at finitely many points and have finite determinant value group, and the irreducible cuspidal automorphic representations of $GL_n(\mathbb{A}_F)$. Furthermore, this correspondence matches Frobenius eigenvalues with Hecke eigenvalues (with multiplicities), which means the L-functions coincide.

If you want something explicit, think of your preferred smooth projective curve defined by $y^2 = x(x-1)(x-\lambda)$, for some appropriate $\lambda \in \mathbb{Z}$. Do not think about $\mathbb{P}^1$ too hard, since the correspondence is trivial in this case. The function field $F$ is the field of globally defined rational (or "meromorphic") functions on $X$. For $X = \mathbb{P}^1$, the function field would be just $\mathbb{F}_q(t)$, and for any curve not isomorphic to $\mathbb{P}^1$, the function field is a finite extension of $\mathbb{F}_q(t)$.

The Galois side

An $n$-dimensional Galois representation is just a representation of $Gal(\overline{F}/F)$ (the absolute Galois group of the function field) in an $n$-dimensional vector space, and l-adic means it is a vector space over some finite extension field of $\mathbb{Q}_\ell$. Since the absolute Galois group carries a natural profinite topology, and an extension field of $\mathbb{Q}_\ell$ is a topological field, the continuity condition is taken with respect to these topologies. To define the ramification condition, we call a Galois representation unramified at a point $x \in X$, if the inertia subgroup $I_{\overline{x}}$ of any lift $\overline{x}$ of $x$ to the maximal unramified cover of $X$

$$I_{\overline{x}} = \{\sigma \in Gal(\overline{F}/F) | \sigma \overline{x} = \overline{x} \text{ and } \overline{\sigma} \in Gal(\kappa(\overline{x})/\kappa(x)) \text{ is trivial}\}$$

acts trivially, where $\kappa(x)$ is the residue field of $X$ at $x$.

If this ramification condition troubles you, think of it by what it provides: We want to look at the geometric Frobenius element $Fr$ in

$$Gal(\kappa(\overline{x})/\kappa(x)) = Gal(\overline{F_{q^n}}/F_{q^n}) = \hat{\mathbb{Z}}$$

($\hat{\mathbb{Z}}$ is the profinite completion of the integers), which is a topological generator inverse to $z \mapsto z^{q^n}$ in $Gal(\overline{F_{q^n}}/F_{q^n})$. This geometric Frobenius can be lifted arbitrarily to an element of the decomposition group

$$D_{\overline{x}} = \{\sigma \in Gal(\overline{F}/F) | \sigma \overline{x} = \overline{x}\}$$

and this lift is unique up to elements of the inertia subgroup $I_{\overline{x}}$.
If a representation of the Galois group becomes trivial upon restriction to the inertia subgroup of a point, there is an action of the lifted geometric Frobenius, unique up to conjugation. All conjugation invariants are thus unique, like the eigenvalues, and in particular the trace of the Frobenius at the point.
So, if a representation is unramified except at finitely many points, there are Frobenius eigenvalues at all but finitely many points.

The Automorphic Side

The Adèles are the ring $\mathbb{A}_F = \lim_{\rightarrow S} \mathbb{A}_F(S)$, where $S \subset X$ is any finite set of points, and the $S$-Adèles $\mathbb{A}_F(S)$ are defined as

$$\mathbb{A}_F(S) = \left(\prod_{x \in S} F_x\right) \times \left( \prod_{x \in X\setminus S} \mathcal{O}_x \right)$$

with $F_x$ the complete local field obtained by completing the field $F$ at the valuation from $x$ and $\mathcal{O}_x$ the (complete) ring of integers in this local field.
The inductive limit in the definition of $\mathbb{A}_F$ goes over the inductive system of all finite sets of points of $X$, ordered by inclusion. The result has a natural topology coming from the inductive-limit-topology of the $S$-Adèles and the $S$-Adèles carry the natural product topology.
There is a more down-to-earth description of the ring $\mathbb{A}_F$ (called restricted product): Its elements are those elements of the big product $\prod_{x \in X} F_x$ which have all but finitely many components in $\mathcal{O}_x$ (but the components are not fixed for the whole set, only for a single element, hence the inductive limit in the former description).
The ring $\prod_{x \in X} \mathcal{O}_x$ is an open compact subgroup of $\mathbb{A}_F$.

By the way, I asked a question about descriptions of the Adèles on MO, and got several good answers.

The group $GL_n(\mathbb{A}_F)$ is defined as usually to be the group of invertible $n\times n$-matrices whose entries are in the Adèle ring $\mathbb{A}_F$. It has a natural topology, coming from the vector space topology of $n\times n$-matrices over the topological ring $\mathbb{A}_F$, making $GL_n(\mathbb{A}_F)$ a locally compact topological group with open compact subgroup $K := GL_n(\prod_{x \in X} \mathcal{O}_x)$.

The irreducible cuspidal automorphic representations of $GL_n(\mathbb{A}_F)$ are not defined as irreducible representations of this group with some extra conditions, but more conveniently by introducing the space of automorphic functions, one special representation called right-regular representation, and then the irreducible cuspidal automorphic representations are defined to be the irreducible components in the direct sum decomposition of the right-regular representation.
So, we will explain automorphic functions.

Denote by $\mathcal{C}(GL_n(F)\backslash GL_n(\mathbb{A}_F))$ the space of all locally constant functions $f : GL_n(F)\backslash GL_n(\mathbb{A}_F) \to \mathbb{C}$ satisfying these conditions:

• K-finiteness: the right translates of $f$ under the operation of $K$ span a finite dimensional vector space,
• Central character: $f(gz) = f(g)$ for all $g \in GL_n(\mathbb{A}_F)$ and all $z \in GL_n(\mathbb{A}_F)$ in the center (which consists of diagonal matrices with unit entries)
• Cuspidality: Given a maximal parabolic subgroup $P$ of $GL_n$ with unipotent radical $U$, this integral over the quotient $Q := U(F)\backslash P(\mathbb{A}_F)$ for all $g \in GL_n(\mathbb{A}_F)$ vanishes:
  

  $$\int_Q f(ug) du = 0.$$

The names "central character" and "cuspidality" actually make sense, but I won't explain this in too much detail here. If you don't know what a parabolic subgroup or a unipotent radical is, a correct description for $GL_n$ is: there exists a decomposition $n = n_1 + n_2 + \cdots n_k$ of $n$ and the block matrices with blocks of size $n_1, n_2,...,n_k$ on the diagonal, arbitrary entries in the upper right part and zero in the lower left part, form the parabolic subgroup, and the unipotent radical consists of those matrices where the diagonal blocks are all the identity matrix. Actually, I have to say that every parabolic subgroup is just isomorphic to one of these (standard) parabolic subgroups, but I guess you're already happy with an explicit description.

Now it's a theorem of Piatetski-Shapiro and Shalika that the natural right action of $GL_n(\mathbb{A}_F)$ on the space $\mathcal{C}(GL_n(F)\backslash GL_n(\mathbb{A}_F))$ decomposes in a direct sum of irreducible representations, all with multiplicity one (this theorem requires and thus justifies the strange conditions for automorphic functions). Each such irreducible direct summand is called irreducible cuspidal automorphic representation.

While this description of the automorphic side can be quite confusing and messy on first sight, I hope that I can convince you that the geometric translation is more natural to look at, but that has to wait for the next blog post.

Frobenius and Hecke eigenvalues, L-functions

In the geometric translation, the matching condition on Frobenius and Hecke eigenvalues is very intricate and I decided not to comment on it at all.

In the function field case, by "Frobenius eigenvalues" on the Galois side, I mean the eigenvalues of the Frobenius lifts at the unramified points. The "Hecke eigenvalues" on the automorphic side are somewhat more complicated, as they are given by the eigenvalues of certain Hecke operators that span the spherical Hecke algebra, defined as convolution algebra over the automorphic functions. The Hecke operators whose eigenvalues are meant are the characteristic functions of certain double-cosets coming from the Bruhat decomposition. If this description is too sketchy, have a look at Frenkel's nice article now!

Given these eigenvalues at each place, one can assemble them into an L-function by an Euler product. The matching can now be expressed by stating that the L-function from a Galois representation coincides with the L-function from an automorphic representation. Some people like to see the whole Langlands business as extension of the comparison of L-functions.