Langlands Programs: Learn What They Are & Why You're Curious

[NoodleDurh]

What are these Langlands Programs. I am curious.

 

[Therodre]

Hi,
It is a vast and deep generalisation of class field theory. But answering to such a broad question would require a long and possibly technical message.
Have you tried wiki?
Can you narrow your questions about the langlands program a tad?

Here is the gist of the Langlands Program (well, what i know about it).

The general idea is to understand the "structure" of $$G_Q=Gal(\overline{\mathbb{Q}}/\mathbb{Q})$$ (or more generally of absolute galois group of number fields, and their local or geometric cousins, but let's stick to Q for starters). What is meant here by structure is not obvious. What we look for is not, say a presenation of G_Q, or the nulber of topological generators or things like that.

One thing that is important to understand is that rigorously speaking G_Q is not well defined. You need to choose and algebraic closure of Q to be able to construct G_Q, and even though all algebraic closures will be isomorphic, if you choose another one, the new G_Q will differ by a conjugation. So the best thing we can master is an understanding of the conjugacy class of G_Q. A nice way to do this, the so called tannakian approch, is to study the representation classes of G_Q, beause they will not vary under conjugation.

Now, I'm not going to describe precisely what is meant by that "structure", but the (vague) idea is to relate this structure to the "arithmetic" of the extensions of Q.
It is a basic fact that you can classify the finite galois extensions of Q (or any number field) by the data of the primes the split in the extension, but what is important is to understand, how can you describe the extension corresponding to a splitting set of primes, and the "other direction" if you have a finite galois extension how to find the primes that split in it.
That's roughly what is meant by relating the arithmetic structure of a number field to its galois extensions. Of course the easiest way to study the galois extensions of Q is via it galois group.

The answer to these questions when you look at abelian galois extension is provided by class field theory. The Langlands program should be able to answer these questions in a more general context.

 

[NoodleDurh]

Okay, you pretty much answered my question. But a follow up question, So if I am going to learn about these langlands programs. What do I need to learn.

 

[jgens]

So if I am going to learn about these langlands programs. What do I need to learn.

I asked one of my professors a similar question a couple years back and I will paraphrase the answer he gave me:

Essentially there is no single collection of techniques one ought to learn for Langlands and mathematics from widely different fields has been brought into tackle different cases of the conjecture. To get your foot in the door, however, you should learn the following things:

1. Algebraic and Analytic Number Theory: The Langlands Conjectures are (in an appropriate sense) wide-reaching extensions of class field theory. Since this belongs to number theory one ought to learn that much at least.
2. Algebraic Geometry: Many of the function fields that arise in the Langlands program come from curves over finite fields, and since this belongs in the realm of algebraic geometry, it certainly helps to have some knowledge here.
3. Automorphic and Modular Forms: (Forgot what was mentioned here)
4. Representation Theory: (Forgot what was mentioned here)

Once you have these prerequisites down you can start learning more focused techniques to handle specific cases of the conjectures.
​

So while I forget why exactly representation theory and automorphic/modular forms are crucial in the program, hopefully that paraphrase helps some. Knapp also has a paper listing out the prerequisites on his website and it might be worthwhile to read that sometime. Here is the link: http://www.math.sunysb.edu/~aknapp/pdf-files/china1.pdf

 

[mgkii]

Hi @NoodleDurh

I can strongly recommend an audiobook I've just listened to by Edward Frenkel called Love and Math - it's specifically about trying to explain the Langlands program (and his part in it). It's about 18 hours long and it's a fast jog - there's so much breadth to the program it's staggering. The audiobook worked for me, but I have to say on reflection the good old paper version would have been easier; some of the concepts are so subtle you have to read the same paragraph multiple times!

It's a great book and Edward Frenkel is not only clearly a genuis, but also has the gift of being able to explain mind bending concepts!