Questions tagged [langlands-conjectures]
Higher reciprocity laws
368
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The first odd degree-2 Artin representation for which the Artin conjecture was proved
At the DeKalb conference on Hilbert's problems, John Tate gave a masterly survey of Problem 9, the General Reciprocity Law. He ends with a discussion of the Langlands Programme, especially the case ...
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Shtukas for $\mathrm{Spec}\,\mathbf{Z}$
This is a very soft and speculative question. Please feel free to downvote, close or delete it.
Studying the cohomology of moduli spaces of shtukas, Drinfeld proved the Langlands program for $\mathrm{...
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Why is Langlands functoriality usually related with period integral in a third group?
In the introduction of "PERIODS OF AUTOMORPHIC FORMS "by
HERVE JACQUET, EREZ LAPID, and JONATHAN ROGAWSKI, they said
"In many cases, it should be possible to characterize the $H$-distinguished ...
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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) ...
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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 ...
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History of points of view on Eisenstein series
What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?
There are, as far as I know, two major perspectives on what Eisenstein series are. The first is in ...
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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 ...
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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 ...
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Request for errata for Automorphic Forms on GL(2)
Edit (7/21/2014): We have finished proofreading Jacquet-Langlands and posted it to Robert Langlands's publications site. If you would like a copy, please download it from here:
http://publications....
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Where stands functoriality in 2017?
In 2002, R. Langlands put forward a new strategy to prove the general functoriality conjecture in the Beyond endoscopy paper. The main purpose of this strategy is to detecting the automorphic ...
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Residue of Eisenstein Series on GL(n)
Reference: Mœglin, C.; Waldspurger, J.-L.: Le spectre residuel de GL(n)
On GL($n$), the result of Waldspurger shows that if an automorphic representation $\pi$ is non-cuspidal and in the discrete ...
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Is there a theory of Maaß forms over finite fields ?
Here is a somewhat naïve question which must have occurred to many people, so it would be nice to record here the attempts at an answer :
Is there a theory of Maaß forms over $\overline{\mathbf{F}}_p$...
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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 $G^\vee$ ...
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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 ...
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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) = \int\limits_{G_\...
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Langlands program vs Shimura-Taniyama-Weil conjecture
Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...
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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 ...
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536
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Endoscopic group that is not a subgroup
The question is a very little more than what's in the title. It is easy (for some values of ‘easy’) to produce examples of endoscopic groups that are not subgroups. When I asked a colleague, he ...
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standard zero free region of automorphic L-function on GL(N)
Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$.
What's the standard zero-free region for $L(s,\pi)$? any ...
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Langlands dual group in math vs. Goddard-Nyuts-Olive dual group in physics
Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.
Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.
Are ...
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Why is the Langlands dual group always taken over $\mathbb{C}$?
Whenever I read a statement of the Langlands conjectures for a reductive group $G$, they are formulated in terms of the Langlands dual group, which is essentially the reductive group over $\mathbb{C}$ ...
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A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$
In the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(...
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Self-dual automorphic forms on $GL(4)$
As is known among experts, all self-dual automorphic forms on $GL(3)$ come from symmetric square lifts from $GL(2)$. You can find this in Ramakrishnan (http://www.math.caltech.edu/~dinakar/papers/...
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Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?
The Generalized Lindelof Hypothesis says that for the $L$-function of an automorphic form we have
$$L(1/2+it)\ll Q(t)^{\epsilon}$$
for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $...
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Why are characters orthogonal to cusp forms?
Let $G = \operatorname{GL}_2$, and let $V = L^2(Z(\mathbb A)G(\mathbb Q) \backslash G(\mathbb A),\omega)$, for $\omega$ a character of the ideles $\mathbb A^{\ast}$, identified with a central ...
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The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)
When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
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Rankin-Selberg for Maass form GL(3)xGL(2)
Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character).
Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume ...
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Symmetric Fifth Power Lift of GL(2) Automorphic Form
Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that
$$L(s, \pi, Sym^m)$...
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Langlands program beyond CM fields?
I apologize since this is a quite vague question. And I am personally at an expert in these fields at all.
It seems to me that there are two main directions of the Langlands program, namely, ...
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Is the nonvanishing of Langlands L-functions at $s=1$ conjectured?
Suppose $G$ is a semisimple algebraic group over the rational numbers, $\pi$ is a cuspidal automorphic representation of $G$, and $r: \widehat{G}(\mathbf{C}) \rightarrow \mathrm{GL}_N(\mathbf{C})$ is ...
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Absolute convergence of Rankin–Selberg series
Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
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L-packets in the local Langlands correspondence: why finite sets?
Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local ...
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Conductor of a representation of a $p$-adic group
Let $G$ be a connected split reductive group over $\mathbb{Z}$. Let $F$ be a local non-Archimedean field. Let $\rho$ be an irreducible smooth representation of $G(F)$. How does one define the ...
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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 ...
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Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$
Let $G = \operatorname{GL}_n(F)$ for a $p$-adic field $F$, and let $G_D = \operatorname{SL}_n(F)$. I am wondering if there is a connection between irreducible, admissible representations of $G$ and ...
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Simplest case of Langlands-Shahidi method
I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function.
Could the constant term of $\mathrm{GL}(2)$-Eisenstein series be ...
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Bernstein–Zelevinsky classification for classical groups
Bernstein and Zelevinsky classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations. The irreducible modules are ...
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How should the local Langlands correspondence for general reductive groups take into account different inner forms?
Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local ...
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How to deduce Bernstein-Zelevinsky classification from the Langlands one
I am trying to understand the Langlands classification. To that end, I am trying to find how I could deduce the Bernstein-Zelevinsky classifcation from the second description of the Langlands ...
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On a remark of Langlands
I'm been wondering about this for a while and hope someone can enlighten me.
In this interview of Robert Langlands's from 2010, on pg 21 (Question 8) he states "At one point, when fairly young, I ...
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Results conditional on Langland's conjectures?
I know in number theorythere are loads of conditional results, dependant on RH for instance. On the other hand, Langland's programme is supposed to provide some understanding of the absolute Galois ...
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What can we say about the Local Langlands Correspondence for GL_n without using Bernstein-Zelevinski?
I have two specific questions regarding the LLC for $GL_n$, and in particular, what we can say about the conjecture if we don't have the ideas of Bernstein and Zelevinski, which reduce the problem to ...
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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 ...
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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 ...
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Contemporary introduction to Godement-Jacquet "Zeta functions of simple algebras"
The question is in the title: The book
Godement-Jacquet "Zeta functions of simple algebras"
is from 1971. Has there ever been a textbook introduction to this material, or at least part of it?...
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Classification of quasi-split unitary groups
Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...
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Examples of function fields Langlands for small genus (<= 2)
See Edward Frenkel's article "Lectures on the Langlands program and conformal field theory" for an exposition of the function fields Langlands correspondence (now a theorem of Drinfel'd, L.Lafforgue &...
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Non-existence of "higher" Artin map
So rank $1$ local Langlands is special in as that it is given by the Artin map
$$\text{GL}_1(K)\to G_K^{ab},$$
whereas in the higher rank (to the best of my knowledge) there doesn't exist a map
$$\...
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Dictionary of arithmetic symmetries and Langlands
To a number theorist automorphic forms appear to be adelic point-counting generating functions for arithmetic schemes. This is what the conjectured equality of their $L$ functions tells us.
The fact ...
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Status of local Langlands conjecture over positive characteristic
I am interested to know what the status of Local Langlands Conjectures in positive characteristic is? By a positive characteristic local field, I mean a field of the form $\mathbb{F}_q((t))$.
A nice ...