Explaining the difference between the number theoretic Langlands program and geometric Langlands program to a graduate student.

Question

I am a graduate student who just took a course introducing some notions in algebraic number theory and algebraic geometry (officially, it was a course on an introduction to the Langlands program). Almost every lecture, the professor spoke about some geometric analog of some field-theoretic object / algebraic object. Can anyone provide a general description of the relationship between the classical Langlands program and the geometric Langlands program? If the difference is very technical and out of the reach of a novice, could you direct me to a resource? Thanks.

Answer 1

The basic idea behind the Langlands program, very roughly and imprecisely, is to associate $n$-dimensional Galois representations (algebraic objects) to "automorphic representations" of GL($n$) (more analytic objects--these are infinite dimensional for $n > 1$). (There are other aspects to the Langlands program, but this is in my mind the most fundamental one.) Both kinds of these representations are defined over a global field, which is classically a number field, but could be also a function field.

(There are also analogues over local fields, just as one has many analogies between local and global fields in classical number theory, e.g., local and global class field theory. In fact the $n=1$ case of the Langlands correspondence may be viewed as a restatement of class field theory.)

The function field version of the Langlands correspondence can be interpreted more directly in terms of geometry, e.g., curves over finite fields. As I understand it (I am no expert), the geometric Langlands correspondence is an attempt to understand what the geometric analogue should be if one works with curves over the complex numbers, rather than a finite field.

The details are rather technical. There is a dictionary on nLab that may be of cursory interest, though it requires significant background to really understand. If you want to spend some time to learn about these things, a couple of references, respectively for classical and geometric Langlands, are Gelbart's "An elementary introduction to the Langlands program" and Gaitsgory's "Informal introduction to geometric Langlands" in the Introduction to the Langlands Program volume (edited by Berstein and Gelbart).