# Questions tagged [langlands-conjectures]

Higher reciprocity laws

362
questions

2
votes

0
answers

60
views

### Local systems on $\mathbb P^1$ and on the formal punctured disc

Consider the projective curve $\mathbb P^1$ over a finite field $k$.
Consider $\ell$-adic local systems $E$ on $\mathbb P^1\backslash \{0,\infty\}$ such that
a) $E$ is tame at $\infty$
b) The ...

7
votes

0
answers

119
views

### Non-abelian ray class fields for local fields

Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...

1
vote

0
answers

96
views

### Langlands dual of torus

Let $T$ be a split torus over a field $k$. Then the dual torus $\hat{T}$ is defined to be the unique torus such that
$$
X_*(T)=X^*(\hat{T}),
$$
where the left hand side is the cocharacter lattice of $...

1
vote

1
answer

170
views

### Non-split extension of representations of $\mathrm{GL}_2$ and $\mathrm{Hom}$

Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...

5
votes

2
answers

453
views

### Non-trivial extension of representations have same central character

Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...

3
votes

0
answers

263
views

### Can you prove and/or generalize this formula involving the Möbius function at n = square free numbers for elliptic curve related sequence in the OEIS?

Let $g(n)$ be the Dirichlet inverse of the Euler totient function:
$$g(n) = \sum\limits_{d|n} d \cdot \mu(d)$$
and let $f(x,y)$ be the elliptic equation:
$$f(x,y)=x^3 - x^2 - y^2 - y$$
Show that the ...

5
votes

1
answer

184
views

### Two different local Langlands parameters for quadratic extension

Let $E/F$ be a quadratic extension of nonarchimedean local field, and let $G$ be a reductive group over $E$, and $G(E)$ the $E$-points of $G$. (For my question, one may just assume $G=\operatorname{GL}...

3
votes

0
answers

84
views

### Langlands parameters and Weyl group actions

Let $F$ be a $p$-adic field and $\mathbf{G}$ a connected reductive group over $F$, assumed to be quasi-split. Let $\mathbf{T}$ be a maximal split torus of $\mathbf{G}$ and $\mathbf{P}=\mathbf{M}\...

6
votes

0
answers

225
views

### What, if anything, do we hope and expect to understand about (classical) Galois groups?

I was reading Franz Lemmermeyer's introduction to Fermat's Last and Wiles' Theorem, where he states
Galois representations $\rho_p : G_\mathbb Q\rightarrow GL_2(\mathbb Z_p)$ are used for studying ...

6
votes

0
answers

305
views

### Can Langlands correpondence be restated using topos?

Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions.
Laurent Lafforgue applying Olivia Caramello thesis described in ...

1
vote

2
answers

165
views

### How to assign the $L$- and $A$-parameters for the trivial representation

Let $G$ be a (split) reductive group over a $p$-adic field $F$, and $\mathbf{1}$ the trivial representation of $G(F)$. Under the (conjectural) Langlands correspondence, this should correspond to an $L$...

2
votes

1
answer

227
views

### Non-continuous group homomorphism from p-adic field to C*

Let $F$ be a p-adic field. A character on $F^\times$ is defined as a continuous group homomorphism $F^\times\longrightarrow\mathbb{C}^\times$. But is there any way to construct a non-continuous group ...

2
votes

1
answer

77
views

### Image of the intertwining operator for GL(2) is $K$-invariant at the "pole" $s=1$

I am taking a look at the residues of Eisenstein series and have a question about a local computation. Let $k$ be a local field, $G = \operatorname{GL}_2(k)$, and $P$ (resp. $K$) the standard ...

3
votes

1
answer

183
views

### Parabolic induction, and tensoring (Iwahori/affine) Hecke algebras

In several places, such as [1] and [2], it seems to be implicitly known that (normalized) parabolic induction corresponds to tensoring affine Hecke algebras. I say implicitly because both [1] and [2] ...

1
vote

1
answer

103
views

### Continuity of central character [closed]

Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group ...

4
votes

0
answers

168
views

### Several L-functions but one Galois representation: How to choose

Let $\mathbf{G}$ be a reductive group which enjoys all the nice properties a reducive group can dream of. Fix $(\mathbf{G},X)$ a Shimura datum associated with it and assume that if $K\leq\mathbf{G} $ ...

3
votes

0
answers

184
views

### Correspondence between motives and automorphic representations

What I know:
I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...

3
votes

0
answers

293
views

### References on $p$-adic Langlands

As a student with a background in Algebraic Geometry (up to chapter 3 of Hartshorne) and basic Algebraic Number Theory, where should I begin learning about the $p$-adic Langlands program? What are the ...

3
votes

1
answer

129
views

### Residue of a local $\gamma$-factor and its relation with adjoint $\gamma$-factor

I met the following relation (if my understanding is correct) of local $\gamma$-factors when I was reading Hiraga-Ichino-Ikeda's paper "Formal Degrees and Adjoint $\gamma$-Factors": Let $F$ ...

2
votes

0
answers

214
views

### Compatibility of Lefschetz formula and categorical local Langlands correspondence

Lefschetz-Verdied formula is formulated for etale sheaves and coherent sheaves in SGA 5 Ⅲ Theorem 4.4 for noetherian schemes.
My questions are that
1.Do we formulate the formula for objects and ...

3
votes

1
answer

110
views

### Modular forms on central division algebra of degree $\ge 3$

I just learned from some online sources including Buzzard's note and Emerton's answer on this MO question about quaternionic modular forms and explicit version of Jacquet-Langlands correspondence in ...

4
votes

1
answer

568
views

### Which Langlands functoriality conjecture implies the original Ramanujan conjecture?

I heard that the Langlands functoriality conjecture implies the Ramanujan conjecture for $GL(2)$. especially for the Maass form.
There are various versions of the Langlands functoriality concerning to ...

3
votes

1
answer

303
views

### $L$-parameters and parabolic induction

I apologize in advance if the answer to this question is well-known to experts.
So let $F$ be a $p$-adic field, and $G$ a reductive group over $F$. Let $P$ be an $F$-parabolic subgroup of $G$ and $M$ ...

2
votes

1
answer

138
views

### Basic results concerning the intertwining operator in the $\mathrm{SL}_2$ case

I am reading [Ikeda, Tamotsu, On the location of poles of the triple L-functions]. On page 194, the author recalled some known results concerning $\operatorname{SL}_2$. I would like to know any ...

2
votes

0
answers

139
views

### Meaning of the meromorphic continuation of intertwining operators

I am trying to make sure the meaning of the meromorphic continuation of the intertwining operators.
Assume we deal with a non-Archimedean field $F$ and just consider $G= SL_2$, for simplicity. We fix ...

4
votes

0
answers

111
views

### $L^2$-spectrum versus automorphic discrete spectrum

Let $G$ be a classical group defined over a number field $F$.
In his monumental book, (https://www.ams.org/books/coll/061/coll061-endmatter.pdf) Arthur described a spectral decomposition of $L_{disc}^...

6
votes

1
answer

484
views

### Langlands-Shahidi method in classical language

The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and ...

3
votes

1
answer

160
views

### Frobenius-Schur indicator of a self-dual L-parameter

Let $F$ be a non-archimedean field and let $\pi$ be a self-dual supercuspidal representation of $\mathrm{GL}_n(F)$ (which, by a result of Adler exists only when $n=1$ or $n$ is even). Then, under LLC ...

1
vote

0
answers

85
views

### Pure, residual, 2-dimensional, semisimple $G_{\mathbb{Q}}$ representations

Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field ...

4
votes

0
answers

137
views

### On decomposition of the space of automorphic forms (via central characters)

Let $F$ be a number field and $\mathbb{A}$ be its adele. For simplicity, we assume $G$ is a connected reductive group.
Given a unitary central character $\chi: Z_{G}(F)\backslash Z_{G}(\mathbb{A}) \to ...

4
votes

0
answers

155
views

### What are the modularity conjectures for Artin motives?

Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...

5
votes

1
answer

534
views

### Understand the $p$-adic local Langlands correspondence with examples

Let $\rho:G_{\mathbb{Q}}\rightarrow \mathbf{Gl}_{n}(\mathbb{Q}_{p})$. I would like to understand in depth why the local Langlands correspondence for $\rho_{|\mathbb{Q}_{p}}$ must consider $p$-adic ...

5
votes

1
answer

776
views

### Can Taniyama-Shimura conjecture be generalized to curves of higher genus (within Langlands framework)?

The Shimura-Taniyama-Weil conjecture asserts that if E is an elliptic curve over Q, then there is an integer N ≥ 1 and a weight-two cusp form f of level N, normalized so that a1(f) = 1, such that ap(E)...

2
votes

0
answers

196
views

### Basic question on the Langlands conjectures for $GL_n$ over global field of positive characteristic

My field is far from the Langlands conjectures. I am just trying to understand some basic ideas.
At the moment I am interested in a global field $K$ of positive characteristic and the group $G=GL_n$. ...

4
votes

1
answer

158
views

### On a theorem of Bernstein-Zelevinsky regarding supercuspidal resentations

Let $G$ be a $p$-adic reductive group and $\pi$ an irreducible non-supercuspidal representation. Then there exist a parbaolic subgroup $P=MN$ and a supercuspidal representation of $M$ such that $\pi$ ...

8
votes

1
answer

309
views

### 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 ...

17
votes

1
answer

1k
views

### How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?

In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...

5
votes

0
answers

182
views

### Finite coefficients Langlands for function fields

Do we have a finite coefficients Langlands correspondence for function fields? By which I mean a bijection between Galois representations $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)...

2
votes

0
answers

168
views

### Why are they called reductive groups? [duplicate]

The reductive groups play a central role in the Langlands correspondence. Why are these groups called reductive? Does this name suggest something conceptual about these groups?

1
vote

1
answer

293
views

### From Galois representations to automorphic forms for $\mathfrak{sl}_2$ (via Drinfeld's shtukas)

Drinfeld-Lafforgue have proven function fields Langlands conjectures in type A: see https://arxiv.org/pdf/math/0212417.pdf (Laumon's survey in English), https://arxiv.org/pdf/math/0212399.pdf (...

4
votes

0
answers

146
views

### Understanding Shimura correspondence in context of Langlands functoriality

Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...

7
votes

1
answer

416
views

### 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
$$\...

19
votes

1
answer

2k
views

### What are "branes", and why do they form a category?

I've been trying to read Kapustin–Witten - Electric–Magnetic Duality And The Geometric Langlands Program recently, as someone whose mathematical interests are in the Langlands program. I have some ...

5
votes

1
answer

302
views

### modularity lifting theorems for non-compact unitary groups

I am reading David Geraghty's paper, 'Modularity lifting theorems for ordinary Galois representations'(https://link.springer.com/article/10.1007/s00208-018-1742-4) and I have a related question, which,...

5
votes

1
answer

211
views

### Motive associated to a cuspidal representation of $GSp_{4}$

In the paper by L.Clozel in this book (a French text), there is this conjecture (conjecture 4.5 p139)
Conjecture: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote ...

10
votes

1
answer

383
views

### 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 ...

4
votes

1
answer

127
views

### The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$

I'm reading James Arthur's notes on the trace formula and am confused on a point on pages 65 and 66. For $G$ a reductive group over $\mathbb Q$ we are going over the decomposition of the space $L^2(G(...

15
votes

2
answers

2k
views

### Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh

In upcoming work of Ben-Zvi-Sakellaridis-Venkatesh, (see for instance these notes or this lecture) some important aspects of the Langlands correspondence are stated in the language of topological ...

6
votes

0
answers

528
views

### Affine GIT quotients and the excursion algebra in Fargues–Scholze

Some background:
Let us fix a non-archimedean local field $E$ with residue characteristic $p$, and let $G$ be some connected reductive group over $E$. In [FS, §VIII.1.1] the authors define a moduli ...

4
votes

0
answers

228
views

### The link between Satake parameter and Godement-Jacquet L-function of an automorphic representation of $GL_{n}$

Origin of the question: I'm reading the following survey of K. Martin, more generally I'm looking for the "best way" to define L-function associated to an automorphic representation of a ...