All Questions
Tagged with class-field-theory langlands-conjectures
10 questions
44
votes
2
answers
7k
views
Why is Class Field Theory the same as Langlands for GL_1?
I've heard many people say that class field theory is the same as the Langlands conjectures for GL_1 (and more specifically, that local Langlands for GL_1 is the same as local class field theory). ...
- 15.4k
31
votes
2
answers
3k
views
Elementary Aspects of Galois Deformation
Galois deformations are an important tool in Wiles' arsenal
for proving FLT. Are there any more elementary aspects (I'm
thinking of 1-dimensional Galois representations attached to
number fields) ...
- 32.6k
23
votes
1
answer
2k
views
Any open Langlands Conjectures for GL_1?
Are there any general conjectures/properties (in the Langlands Program) for automorphic representations of $GL_n$ which are still open for $n=1$?
- 1,629
14
votes
5
answers
3k
views
What is the "reason" for modularity results?
The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? .
I ...
9
votes
1
answer
322
views
A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$
In the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(...
- 311
7
votes
0
answers
157
views
Non-abelian ray class fields for local fields
Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...
- 1,116
4
votes
0
answers
110
views
Uniqueness of class field theory map
Let $F$ be a local field of characteristic 0. The main theorems in local class field theory can be summarized by the existence of a group $W_F$ and a map
$$
\phi_F:W_F\to W_F^\mathrm{ab}\simeq F^\...
- 361
2
votes
1
answer
232
views
A certain idele class character
Let $E/K$ be a cubic extension of number fields, $\nu$ be a Grossencharacter of the idele class group $\mathbb{I}_{E}/E^{\ast}$ such that $\nu^2$ is trivial and $\nu$ restricted to the idele class ...
2
votes
0
answers
168
views
Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?
In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...
- 6,180
2
votes
0
answers
119
views
The definition of Langlands' L-function $L(s,\pi,r)$ in the case of $\operatorname{GL}_1$
Let $G$ be a split reductive group over a $p$-adic local field $k$. For $\pi$ an unramified representation of $G(k)$, and $r$ a finite dimensional representation of the L-group $^LG$, Langlands ...
- 6,180