All Questions
Tagged with nt.number-theory rt.representation-theory
473 questions
63
votes
1
answer
4k
views
Feit-Thompson conjecture
The Feit-Thompson conjecture states:
If $p<q$ are primes, then $\frac{q^p-1}{q-1}$ does not divide $\frac{p^q-1}{p-1}$.
On page xiii of these proceedings of a conference at the University of ...
- 26.5k
51
votes
2
answers
4k
views
Which philosophy for reductive groups?
I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be ...
- 5,424
46
votes
1
answer
3k
views
What is the status of Arthur's book?
Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted:
The ...
- 6,039
42
votes
2
answers
8k
views
Current Status on Langlands Program
The Langlands Program was launched almost fifty years ago, and progress has been made gradually, much of it hard earned. Langlands himself wrote a survey on the functoriality conjecture in 1997, Where ...
- 3,799
40
votes
6
answers
4k
views
What motivations for automorphic forms?
Automorphic forms are ubiquitous in modern number theory and stands as a mysterious Graal lying at the intersection of many fields, if not building valuable bridges between them. However, since this ...
36
votes
1
answer
2k
views
Tell me an algebraic integer that isn't an eigenvalue of the sum of two permutations
Can you tell me an algebraic integer, with all archimedean absolute values less than 2, which is not an eigenvalue of $\pi_1 + \pi_2$ for any two permutation matrices $\pi_1,\pi_2$?
Is it ...
- 19.2k
33
votes
5
answers
4k
views
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Apologies in advance if this is obvious.
Edit, 5/31/24: Since this question is getting some ...
- 118k
29
votes
0
answers
1k
views
Linking formulas by Euler, Pólya, Nekrasov-Okounkov
Consider the formal product
$$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$
(a) If $z=2$ then on the one hand we get Euler's
$$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$
on the ...
- 43.2k
28
votes
2
answers
5k
views
Which p-adic numbers are also algebraic?
What is $\mathbb{Q}_p \cap \overline{\mathbb{Q}}$ ?
For instance, we know that $\mathbb{Q}_p$ contains the $p-1$st roots of unity, so we might say that $\mathbb{Q}(\zeta) \subset \mathbb{Q}_p \cap \...
- 1,951
28
votes
2
answers
5k
views
Status of (global) Langlands conjecture for $\mathrm{GL}_2$ over $\mathbb{Q}$
$\DeclareMathOperator\GL{GL}$Apologies if this question has already been dealt with on MO. I am wondering about the status of the global Langlands conjectures for $\GL_2$ over the rational numbers. ...
- 283
27
votes
5
answers
3k
views
Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?
Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
- 1,373
27
votes
1
answer
3k
views
Reconciling Lusztig's results with the Langlands philosophy
Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = \mathrm{...
- 805
26
votes
3
answers
5k
views
Questions about the Bernstein center of a $p$-adic reductive group
Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-...
- 809
26
votes
4
answers
4k
views
Overview of the interplay of Harmonic Analysis and Number Theory
I'm kind of disappointed that the question here was never sharpened.
The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...
25
votes
3
answers
1k
views
what else is in $\prod_{j=1}^n(1+q^j)$?
From time to time, I run into the finite product $\prod_{j=1}^n(1+q^j)$. And, the more it happens, the more fascinated I've become. So, herein, I wish to get help in collecting such results. To give ...
- 43.2k
23
votes
2
answers
3k
views
Why are Tamagawa numbers equal to Pic/Sha?
For a connected algebraic group $G$ over a global field $K$ with adeles $A$, the Tamagawa number of $G$ is the volume of $G(A)/G(K)$. It is conjectured (and often known) to be rational, namely the ...
- 8,717
23
votes
1
answer
1k
views
On an asymptotic formula of Keating and Snaith involving the Riemann zeta function
Keating and Snaith have a famous conjecture on the asymptotics of the
integral $\int_0^T |\zeta(\frac 12+it)|^{2k}\, dt$, where $\zeta$
denotes the Riemann zeta function. See page 510 of the book ...
- 50.8k
20
votes
1
answer
4k
views
Cusp forms and L^2
I am confused about the "bigger picture" when one goes from classical modular forms on $SL_2(\mathbb{Z})$ and its subgroups to automorphic forms (possibly non-holomorphic).
For classical modular ...
- 233
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 ...
- 1,000
20
votes
1
answer
786
views
Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$
This question is out of plain curiosity. The first sentence of Deligne's
Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique $0$ (1984) reads (in rough translation) as ...
- 18.7k
19
votes
3
answers
2k
views
Simple instance illustrating significance of Langlands dual group without getting into the Langlands program?
To a reductive group $G$, one can associate its "Langlands dual" group ${}^L G$. The Langlands dual group is notably important in the Langlands program. But it seems to me that the Langlands ...
- 63.9k
19
votes
2
answers
3k
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 ...
- 41.4k
18
votes
6
answers
2k
views
Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic.
Let $K$ be a non-arch local field (I'm only interested in the char 0 case), let $\mathbb{G}$ be a connected reductive group over $K$ and let $G=\mathbb{G}(K)$. If $V$ is a smooth irreducible complex ...
- 41.4k
18
votes
3
answers
2k
views
What's the status of Arthur's announced classification for GSp(4)?
In "Automorphic representations of GSp(4)" (2004) (see http://www.math.toronto.edu/arthur/), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no ...
- 40.2k
18
votes
1
answer
3k
views
What problem do the adeles solve?
While browsing through some papers, I came across some literature discussing the Arthur-Selberg trace formula. At a conceptual level I think I understand what it is doing, but when I get down to the ...
- 2,392
18
votes
4
answers
2k
views
Origin of symbol *l* for a prime different from a fixed prime?
I've never seen an authoritative explanation for the choice of the lower case letter $\ell$ or $l$ to denote an arbitrary prime different from a given prime $p$. This now has its own LaTeX command \...
- 52.9k
18
votes
1
answer
1k
views
Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$One can compute the (group cohomological) Euler characteristic of $\SL_2(\mathbb{Z})$ via
$$ \chi(\SL_2(\mathbb{Z})) = \chi(\mathbb{Z}/2) \...
- 412
18
votes
1
answer
821
views
Torsion units of the ring $\mathbb{Z}[x]/(1+x+x^2+\cdots+x^{n-1})$
What are the torsion units of the ring $R_n:=\mathbb{Z}[x]/(1+x+x^2+\cdots+x^{n-1})$? Since $x^n = 1$ in $R_n$ it is clear that all elements of the form $\pm x^i$ are torsion units. Is this all of ...
- 183
18
votes
4
answers
621
views
What are immediate applications of the classification of connected reductive groups?
After years of putting it off, I finally sat down, read, and understood the classification of connected reductive groups via root data.
That's a non-trivial theory! I'm hoping that now that I am done ...
- 557
17
votes
5
answers
2k
views
A natural way of thinking of the definition of an Artin $L$-function?
Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be $\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 -...
- 7,062
17
votes
2
answers
2k
views
Weil-Deligne representations: Two monodromy operators?
Let $p$ be a prime number, and $F$ be a finite extension of $Q_p$. To any smooth irreducible representation $\pi$ of $G = Gl_n(F)$ we may associate a sort of ``dual´' representation, called the ...
- 1,190
17
votes
4
answers
2k
views
Where do the real analytic Eisenstein series live?
In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...
17
votes
2
answers
1k
views
Which L-functions are not "Langlands-Shahidi L-functions"?
The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
- 3,799
16
votes
2
answers
1k
views
Etymology of cuspidal representations
In the literature on representation theory of $GL_2(\Bbb F_p)$ and $GL_2(\Bbb Q_p)$, the irreducible representations with trivial Jacquet module are often called "cuspidal" or "supercuspidal". Why are ...
- 263
16
votes
1
answer
1k
views
What is the Twisted Trace Formula?
I am studying the trace formula using "An Introduction to the Trace Formula" by James Arthur. I would like to understand the twisted trace formula, but unfortunately I never came across a good ...
- 161
16
votes
0
answers
605
views
Division fields of abelian varieties over function fields
Let $k$ be a finitely generated field (for example a finite field or a number field) and $K/k$ a finitely generated regular extension with $trdeg(K/k)=1$. Let $A/K$ be a principally polarized abelian ...
- 1,673
15
votes
6
answers
1k
views
Conjugacy for $p$-adic matrices of finite order
$\DeclareMathOperator\GL{GL}$Say $p$ is an odd prime, and take two matrices $A,B\in \GL_n({\mathbb Z}_p)$ of finite order $m$. Is it true that they are conjugate in $\GL_n({\mathbb Z}_p)$ if and only ...
- 5,363
15
votes
3
answers
1k
views
Philosophy behind cohomological representations
For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...
15
votes
3
answers
2k
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What is the intuition behind the definition of cuspidal representations?
Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(...
- 153
15
votes
1
answer
1k
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Local Langlands for $GL(2,\mathbf{C})$ and reducible principal series
My naive picture of the local Langlands correspondence for $GL(2,\mathbf{C})$ is this. The Weil group of $\mathbf{C}$ is canonically $\mathbf{C}^\times$. On the Galois side then we're looking at 2-...
- 829
15
votes
1
answer
910
views
A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$
Qeustion:
Given a Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_\ell$ with an ideal $\mathfrak{g}^O$ and a subalgebra $\mathfrak{h}$,
such that $\mathfrak{g}=\mathfrak{g}^O+\mathfrak{h}$.
Now given a ...
- 178
15
votes
1
answer
954
views
Funktorialität in der Theorie der automorphen Formen
In 2010 Langlands wrote an article with the title Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele. On the IAS website, he says that
This note ... was written ...
- 18.7k
14
votes
8
answers
2k
views
Applications of the idea of deformation in algebraic geometry and other areas?
The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...
14
votes
5
answers
2k
views
What is the Hilbert class field of a cyclotomic field?
In the answers to Qiaochu's post on defining representations of finite groups over the algebraic integers, it came out that which fields a representation of a finite group is defined over might depend ...
- 44.7k
14
votes
2
answers
571
views
Number of d-Calabi-Yau partitions
This problem arises from algebraic geometry/representation theory, see https://arxiv.org/pdf/1409.0668.pdf (chapter 2).
We call a partition $p=[p_1,...,p_n]$ with $2 \leq p_1 \leq p_2 \leq ... \leq ...
- 26.5k
14
votes
2
answers
2k
views
"Purely local" proof of local Langlands
As from this website
http://math.uchicago.edu/~lxiao/workshop_site/
My question is: What does it mean by "purely local"?
Also, I heard about this phrase "purely local" in other problems as well, ...
- 1,503
14
votes
1
answer
506
views
Local-global principle for split extensions of Galois representations
I guess the following is well-known (and probably follows from Chebotarev's density theorem, but I'm not very comfortable with it):
Define some notation:
$K$ a global field,
$G$ the absolute Galois ...
- 5,504
14
votes
1
answer
1k
views
Zeroes of Maass forms
By a Maass form I just mean--maybe a bit loosely--any real analytic $\Bbb C$-valued function $f$ on the upper halfplane $\cal{H}$ which is automorphic of weight $k\in\Bbb Z$ with respect to a discrete ...
- 797
14
votes
2
answers
1k
views
Class groups in dihedral extensions - some sort of Spiegelungssatz?
Let $p$ be an odd prime and let $F/\mathbb{Q}$ be a Galois extension with Galois group $D_{2p}$, let $K$ be the intermediate quadratic extension of $\mathbb{Q}$, and $L$ an intermediate degree $p$ ...
- 13k
14
votes
1
answer
2k
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 $\pi$ ...
- 1,190