Higher reciprocity laws

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11
votes
2answers
726 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 ...
11
votes
1answer
937 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 $d/n$ ...
14
votes
1answer
1k 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 ...
56
votes
7answers
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
1answer
1k 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 ...
23
votes
2answers
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) ...
13
votes
3answers
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 ...
26
votes
3answers
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 ...
57
votes
7answers
5k 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 ...
48
votes
2answers
5k 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 ...
16
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 ...
9
votes
1answer
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
3answers
750 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 examples....
10
votes
1answer
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 ...
29
votes
4answers
3k 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 ...
26
votes
5answers
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 ...