Higher reciprocity laws

**13**

votes

**5**answers

2k 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 ...

**5**

votes

**2**answers

566 views

### Is there for every variety X an abelian variety A such that their 1st l-adic cohomologies are isomorphic?

This question is somewhat inspired by Kevin Buzzard's answer to What is the interpretation of complex multiplication in terms of Langlands? and somewhat from my own curiosity about such topics.
Let ...

**3**

votes

**1**answer

740 views

### What is the interpretation of complex multiplication in terms of Langlands?

I'm trying to interpret things in the following terminology:
Assume the standard conjectures, the existence of the conjectural Langlands group, and anything else you wish.
I assume the following ...

**6**

votes

**1**answer

215 views

### Finite field analogue of representations in same packet have equal central character

In Kevin Buzzard's recent question, a warm up question was: if two automorphic representations are nearly equivalent, then are the central characters of their local components equal?
Working my way ...

**7**

votes

**1**answer

774 views

### How does the conjectural Langlands group fit into the Tannakian point of view?

I've read that one way to formulate the Langlands program is the following:
Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) ...

**18**

votes

**4**answers

1k views

### How badly can strong multiplicity one fail in the theory of automorphic representations?

Let $G$ be a connected reductive group over a global field $k$, and let $\pi=\otimes_w\pi_w$ and $\pi'=\otimes_w\pi'_w$ be two automorphic representations for $G$, where here of course $w$ is ranging ...

**11**

votes

**3**answers

2k views

### What makes Langlands for n=2 easier than Langlands for n>2?

I must confess a priori that I haven't read the proof of Taniyama-Shimura, and that my familiarity with Langlands is at best tangential.
As I understand it Langlands for $n=1$ is class field theory. ...

**2**

votes

**2**answers

601 views

### Are the $L$-functions of $X_0(N)$ automorphic?

This question, like all of my previous questions regarding Langlands, is very naive.
All $g\geq 1$ curves come from quotients of the upper half plane. The curves $X_0(N)$ come from quotients of ...

**3**

votes

**0**answers

327 views

### Motivic interpretation of genus 2 Siegel forms induced by lifts of Maass and Skoruppa

Background: There are several known lifts from integral weight modular forms to Siegel forms of genus 2, among them the Saito-Kurokawa lift. Another lift construction that is important for ...

**12**

votes

**2**answers

1k views

### What is the strongest, most natural, conjectural form of Langlands?

This is inspired by my previous question:
What is the precise relationship between Langlands and Tannakian formalism?
As well as the excellent link that Tom Leinster put in a comment to that thread: ...

**15**

votes

**1**answer

1k views

### What is the precise relationship between Langlands and Tannakian formalism?

As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept!
In any case, I wish ...

**5**

votes

**2**answers

616 views

### What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula?

I have read that there is an elementary way to show that the above mentioned trace fromulas are equivalent in the sense, that each of them can be derived directly from the other. There should exist a ...

**7**

votes

**0**answers

405 views

### Is endoscopy interesting in simply-laced cases?

Let $G$ be a complex algebraic group, and write $Z(g)$ for the centralizer of a semisimple element $g$ in $G$. I will assume $G$ is simply connected, in which case $Z(g)$ is connected.
Let $G^\vee$ ...

**3**

votes

**1**answer

929 views

### Jacquet Langlands correspondance

I have one issue with the Jacquet Langlands correspondance. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this ...

**1**

vote

**1**answer

808 views

### Rapoport-Zink proof of purity of monodromy

Hi,
Does anyone know if the article
"Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindene Zyklen in ungleicher Charakteristik", INvent. Math, 68 (1980)
by ...

**9**

votes

**1**answer

893 views

### Carayol via the trace formula

Hi,
Is there a proof of the result that Carayol proves in "Sur les representations l-adiques..."
using the Langlands-Kottwitz method of comparing the Lefschetz trace formula and the Selberg trace ...

**30**

votes

**2**answers

2k 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). ...

**20**

votes

**3**answers

2k views

### Geometric construction of depth zero local Langlands correspondence

Dear community,
In light of the recent work of DeBacker/Reeder on the depth zero local Langlands correspondence, I was wondering if there is an attempt to "geometrize" the depth zero local Langlands ...

**4**

votes

**1**answer

1k views

### Why is the Arthur trace formula so powerful?

Considering the Arthur trace formula, why are the sort of convolution operators, whose "normalized traces" are given in geometric terms and spectral terms, actually able to distinguish all automorphic ...

**5**

votes

**2**answers

674 views

### Why is the simple trace formula a weaker tool than the Arthur trace formula?

What are some concrete examples of theorems which can be deduced from the Arthur trace formula, which do not follow from the simple trace of Kazdhan and Flicker?
(So I do not mean weaker in the sense ...

**5**

votes

**1**answer

1k views

### Galois representation associated to a modular form is crystalline iff…

I am looking for the reference for the following fact (used, for example, in the proof of theorem 4.4. in Breuil's expose about local-global compatibility at Bourbaki):
For $f$ a modular cuspidal ...

**7**

votes

**3**answers

667 views

### Why do we see SU(2,R) in the Local automorphic Langlands group?

The paper "A note on the automorphic Langlands group" by J. Arthur,
http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf
discusses the mysterious `automorphic Langlands group'. This ...

**5**

votes

**1**answer

871 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, ...

**6**

votes

**1**answer

705 views

### Semisimple Weil-Deligne representations

I've just realized that I don't understand something important and basic about the Weil-Deligne group and its representations. (I'm not very surprised by this).
Following Deligne's article, Section ...

**12**

votes

**1**answer

1k views

### local Langlands and the Jacquet module

Let $G = GL_n(F)$ be the general linear group over a finite extension $F$ of $Q_p$. This question could be posed for a larger class of groups, but let us stay with $Gl_n$ for the moment.
Let $\pi$ ...

**26**

votes

**3**answers

3k views

### What are the pillars of Langlands?

I had previously asked:
Narratives in Modular Curves
Since then, I've read quite a bit more (but not nearly enough) and I have a few follow up questions about the big picture. As you will soon see, ...

**8**

votes

**0**answers

491 views

### questions on Deligne's letter to Piatetski-Shapiro

Now, that the letter is to be found here: http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf
,I'd have a couple of questions concerning it. Actually, I have a problem only with the ...

**12**

votes

**1**answer

872 views

### L-functions and higher-dimensional Eichler-Shimura relation

From what I have been reading I understand that it is a part of the Langlands program to express Zeta-function of a Shimura variety associated to the group G in terms of L-functions attached to G. I ...

**38**

votes

**2**answers

2k views

### Langlands in dimension 2: the Yoshida conjecture

Background:
One prominent part of the Langlands program is the conjecture that
all motives are automorphic.
It is of interest to consider special cases that are more precise, if less
sweeping. ...

**24**

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

1k 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 $S_F$ ...

**14**

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

780 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

489 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 ...

**48**

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

713 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

**1**answer

884 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

**1**answer

984 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

**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

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 ...

**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) ...

**13**

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 ...

**53**

votes

**7**answers

4k 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 ...

**47**

votes

**2**answers

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

**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 ...

**9**

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

726 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**

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 ...