Higher reciprocity laws

learn more… | top users | synonyms

5
votes
3answers
794 views

The historical development of automorphic geometry

Background: Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...
2
votes
2answers
608 views

Elliptic orbital integral

Let $F$ be a local field and $G = GL(n,F)$. Let $f$ be an element $C_c^\infty(G)$. Let $\gamma$ be an elliptic element of $G$ with irreducible characteristic polynomial. What are strategies to ...
5
votes
0answers
517 views

The Shafarevich Conjecture and motivic Langlands stacks.

Hi, I recently learned about an amazing conjecture of Shafarevich (proved by Faltings) about the finiteness of the number of curves of a fixed genus with good reduction outside a finite number of ...
6
votes
1answer
554 views

Followup questions about the relationship between modular forms and motives

It occurs more and more that I ask a question on math stackexchange and then realize that it is more appropriate to mathoverflow. Hopefully this reflects well on myself... In any case, I copy here ...
9
votes
1answer
884 views

Is this a subcase of the fundamental lemma?

Let $F$ be a local field and $G= GL(n,F)$. Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer. The orbital integral is defined as $$ O_\gamma^G( \phi) = ...
1
vote
0answers
341 views

General cohomology groups and motives

Let $X$ be a variety over $\mathbb{Q}$. Let $\mathcal{F}$ be a sheaf on $X$. Then we have an action of $Gal(\mathbb{Q})$ on $H_{et}^i(X,\mathcal{F})$. In certain cases we can say a lot about this ...
3
votes
1answer
852 views

In what way do the Weil Conjectures pertain to Langlands?

For a relative variety $X$ over a ring of integers $O_K$, we can define a zeta function. This zeta function is defined as the product of the zeta functions of the variety specialized to ...
11
votes
2answers
643 views

Insolvable number fields ramified only at one (small) prime

In his first Eilenberg Lecture at Columbia, Benedict Gross says that only recently have we been able to give examples of finite galoisian extensions $K$ of ${\bf Q}$ which are ramified only at $2$ ...
7
votes
2answers
2k views

Consequences of the Langlands program

In the one-dimensional case the Langlands program is equivalent to the class field theory and the two-dimensional case implies the Taniyama-Shimura conjecture. I would like to know are there any ...
2
votes
1answer
519 views

Different cuspidal automorphic representations with same representations at infinity

Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$). Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$. $$K=\Pi_{v<\infty}K_v$$ $K_v$ is ...
4
votes
1answer
571 views

classification of irreducible admissible (g,K)-module for GL(3,R)

classification of irreducible admissible (g,K)-module for GL(3,R) Is there a classification of irreducible admissible (g,K)-module for GL(3,R)? For GL(2,R) we have principal series, discrete series ...
13
votes
5answers
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
2answers
571 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
1answer
763 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
1answer
220 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
1answer
837 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) ...
19
votes
4answers
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
3answers
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
2answers
622 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
0answers
333 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
2answers
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: ...
16
votes
1answer
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 ...
6
votes
2answers
655 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
0answers
412 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
1answer
945 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
1answer
816 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
1answer
902 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
2answers
3k 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
3answers
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
1answer
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 ...
6
votes
2answers
701 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
1answer
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
3answers
684 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
1answer
887 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
1answer
716 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
1answer
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 $\pi$ ...
27
votes
3answers
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
0answers
496 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
1answer
902 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
2answers
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
1answer
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
1answer
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
2answers
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
1answer
783 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
0answers
497 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 ...
50
votes
2answers
4k 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
2answers
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
2answers
718 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
911 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 ...