Higher reciprocity laws

**22**

votes

**1**answer

2k views

### The fundamental lemma and the conjecture of Birch and Swinnerton-Dyer

Here is a rather pathetic question. In a comment on Tim Gower's weblog, I tentatively stated that the fundamental lemma was necessary for the work of Skinner and Urban relating ranks of Selmer groups ...

**13**

votes

**1**answer

917 views

### Galoisian sets and the Langlands programme

Note: I've revised the question just a little bit in the hope of making it easier.
Given an algebraic number field $F$, which we may as well take to be Galois over $\mathbb{Q}$, we denote by $S_F$ ...

**13**

votes

**2**answers

1k views

### Quaternary quadratic forms and Elliptic curves via Langlands?

The content of this note was the topic of a lecture by Günter Harder at the School on Automorphic Forms, Trieste 2000. The actual problem comes from the article
A little bit of number theory by ...

**4**

votes

**1**answer

751 views

### Weil-Deligne representations

Where can I find a complete classification of the $l$-adic Weil-Deligne representations for a local field $F$ of residual characteristic $p$ (with $p$ different from $l$)? Thanks.

**8**

votes

**0**answers

457 views

### Automorphic representations attached to abelian varieties

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose L-function agrees ...

**45**

votes

**2**answers

3k views

### Non-abelian class field theory and fundamental groups

Over the years, I've been somewhat in the habit of asking questions in this vein to experts in the Langlands programme.
As is well known, given an algebraic number field $K$, they propose to replace ...

**16**

votes

**2**answers

1k views

### Uniqueness of local Langlands correspondence for connected reductive groups over real/complex field.

In Langlands' notes "On the classification of irreducible representations of real algebraic groups", available at the Langlands Digital Archive page here, Langlands gives a construction which is now ...

**11**

votes

**2**answers

658 views

### units in distinct division algebras over number fields---are they definitely not isomorphic as abstract groups?

This is really an irrelevant question in the sense that the answer isn't remotely "logically crucial for the Langlands programme" or whatever---it's just something that occurred to me when writing ...

**10**

votes

**1**answer

748 views

### Simple explicit example of local Jacquet-Langlands theorem for inner forms of GL(n), and consequences

This one will be very easy for the experts.
Let $F$ be a nonarch local field, let $n\geq1$ be an integer, choose $0\leq d<n$ and let $D$ be the central simple algebra over $F$ with invariant $d/n$ ...

**12**

votes

**1**answer

872 views

### Characterizing the Local Langlands Correspondence

In the p-adic case, is there any hope for a set of conditions on the local Langlands correspondence which would make it unique? In the case of GL(n) this is provided by L and epsilon factors. For ...

**52**

votes

**7**answers

3k views

### Open project: Let's compute the Fourier expansion of a non-solvable algebraic Maass form.

OK so let's see if I can use MO to explicitly compute an example of something, by getting other people to join in. Sort of "one level up"---often people answer questions here but I'm going to see if I ...

**9**

votes

**1**answer

951 views

### P-adic local Langlands for non-unitary representations?

In Colmez's work on the p-adic local Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$, he works with ${\rm GL}_2(\mathbb{Q}_p)$-representations on $p$-adic Banach spaces which admit an ...

**21**

votes

**2**answers

2k 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) ...

**10**

votes

**3**answers

2k views

### Weil group, Weil-Deligne group scheme and conjectural Langlands group

I was reading a series of article from the Corvallis volume. There are couple of questions which came to my mind:
Why do we need to consider representation of Weil-Deligne group? That is what is an ...

**25**

votes

**3**answers

4k views

### Tools for the Langlands Program?

Hi,
I know this might be a bit vague, but I was wondering what are the hypothetical tools necessary to solve the Langlands conjectures (the original statments or the "geometic" analogue). What I mean ...

**40**

votes

**6**answers

3k views

### What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...

**42**

votes

**2**answers

4k views

### Galoisian sets of prime numbers

The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved ...

**13**

votes

**1**answer

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 ...

**8**

votes

**1**answer

2k views

### The monodromy-weight-, Ramanujan-, Langlands-landscape

The drawing on the last page of Yoshida's notes make me puzzle, perhaps you can help? It shows a "landscape" featuring the monodromy-weight conj., the general Ramanujan-conj., the Langlands ...

**6**

votes

**3**answers

676 views

### examples of admissible representations of $GL_{n}(K)$ over p-adic field

I've been reading about the Langlands program (the paper by Torsten Wedhorn "Local langlands correspondence for GL(n) over p-adic fields, to be precise), and I want to get my hands dirty with ...

**9**

votes

**1**answer

2k views

### Kapranov's analogies

I just wonder about Kapranov's "Analogies between Langlands Correspondence and topological QFT". I would like to read a more detailed exposition and how one turns that analogy into concrete ...

**22**

votes

**4**answers

2k views

### Induction and Coinduction of Representations

I'd like to understand the general framework of induction and coinduction of representations. If G is a finite group and H a subgroup, I know that there is a restriction functor from representations ...

**22**

votes

**5**answers

4k views

### Where stands functoriality in 2009?

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s.
There's a very interesting article by ...