Higher reciprocity laws

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2answers
946 views

Status of (Global) Langlands Conjecture for $GL_2$ over $\mathbb{Q}$

Apologies if this question has already been dealt with on MO. I am wondering about the status of the global Langlands conjectures for $GL_2$ over the rational numbers. How close is humanity to the ...
13
votes
1answer
2k views

What is the current status of the function fields Langlands conjectures?

My question, roughly speaking is, what happened to the function fields Langlands conjecture? I understand around 2000 (or slightly earlier perhaps), Lafforgue proved the function fields Langlands ...
15
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1answer
992 views

Understanding the “idea” behind Langlands

Apologies in advance if this is a bit too simple to ask here, but I think I'm probably more likely to get an answer here than at stackexchange. I've been trying to learn the basics of the Langlands ...
10
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3answers
517 views

What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...
5
votes
1answer
840 views

Langlands conjectures in higher dimensions

Geometric class field theory (curves over a finite field) has been generalized to higher dimensional varieties over a finite field (and other arithmetical fields). Some of the key names here are Lang, ...