Higher reciprocity laws

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14
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1answer
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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
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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
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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
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2answers
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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
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3answers
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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
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3answers
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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
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7answers
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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
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2answers
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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
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1answer
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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
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1answer
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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
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3answers
748 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 ...
10
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1answer
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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 ...
28
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4answers
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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
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5answers
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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 ...