Questions tagged [langlands-program]
In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles.
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Action of the absolute Galois group on the based root datum
Let $G$ be a connected reductive group over a field $k$ with separable closure $k^s$ and $\Gamma=\text{Gal}(k^s/k)$. Pick $T\subset B$ maximal split torus and Borel subgroup for $G_{k^s}$. I am aware ...
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Has a proper version of Bernstein's lectures, written by Karl Rumelhart ever been published as a proper book/article?
I'm specifically refering to the set of ntoes as seen here:
https://people.math.harvard.edu/~gaitsgde/Jerusalem_2010/GradStudentSeminar/p-adic.pdf
This set of notes serves as a great introduction to ...
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Coset representatives for $wPw^{-1} \cap P' \backslash P'$ for parabolic subgroups $P'$ and $P$?
I'm going through the calculation of the constant term of Eisenstein series in Moeglin and Waldsburger's Spectral Decomposition and Eisenstein series book (section II.1.7), and am confused on a small ...
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Is there an error in this Langlands Program video by Quanta Magazine?
Recently, I was watching Quanta Magazine's The Biggest Project in Modern Mathematics YouTube video. I was thrown off by the expansion of $(1-x)^{24}$ to $(1-24x+276x^2)$ at around timestamp 3:40. Is ...
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The Weil group and maximal tame extension
Let $k$ be a $p$-adic field with residue field $\kappa$ of cardinality $q$ and characteristic $p$.
The Weil group $W(k)$ is the subgroup of $\operatorname{Gal}(\overline{k}/k)$ of automorphisms that ...
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Why is our definition of "smooth" function on a totally disconnected space the "right" definition?
My main interest in this comes from starting to learn about representations of totally disconnected, locally compact topological groups $G$, and their representations. I came across the definition of &...
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What are the prerequisites for studying Automorphic Representation Theory?
I am interesting in getting into Automorphic Representation Theory but am unsure where to start. It seems like there are a lot of prerequisites. What are the main areas (above abstract algebra) that ...
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Reference for Langlands functoriality conjecture view towards classical examples
I want to know if there's any good reference on Langlands functoriality conjecture which provides connection with classical examples. What I have in my mind are followings:
Classical Rankin-Selberg (...
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How do algebraic geometry and the Langlands program compare?
At roughly 12:30-13:00 in this video, Ravi Vakil suggests that algebraic geometry is a sort of unifying theory of mathematics.
In this article, Edward Frenkel describes the Langlands program as “a ...
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Trace of an induced representation on $G(\mathbb A)$
Let $G = \operatorname{GL}_2$, with center $Z$, diagonal matrices $T$, and upper triangular unipotents $N$. Let $K$ be the standard maximal compact subgroup of $G(\mathbb A) = G(\mathbb A_{\mathbb Q})...
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Understanding local L-packets for SL(2,F) for a nonarchimedean field F
The local Langlands correspondence for $GL(2,F)$, for a non-archimedean field $F$, gives a one-to-one correspondence between (equivalence classes of) irreducible admissible representations of $GL(2,F)$...
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Where do automorphic Maass forms come from?
I know the Langlands program for GL2/SL2 gives some hints as to the origins of automorphic representations, and that some cases of the correspondence have been completely classified or reasonably ...
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Which Hecke algebra is used in representation theory?
Which Hecke algebra is used in representation theory or more specifically in the study of Langlands's conjecture ?
From here, the Hecke algebra is constructed from a locally compact topological group ...
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Concrete example of non-abelian class field theory - why Langlands program *is* a non-abelian class field theory?
Abelian class field theory generalizes quadratic reciprocity laws for general number fields with abelian Galois groups, which connects class groups and Galois groups via Artin's reciprocity map. Also, ...
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Links between Langlands program, elliptic curves, and cryptography
There is a link between the Langlands program and elliptic curves as well as a link between elliptic curves and cryptography. I am wondering how a thing in the Langlands program can be translated to ...
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Details in Deligne-Serre "Formes modulaires de poids 1"
There are specific details which I'm a little stuck on in Deligne and Serre's paper on attaching Galois representations to modular forms of weight 1.
In the proof of Lemma 8.3, they use the ...
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poles of local zeta integral in Tate's thesis
Let $F$ be a local field and $\omega : F^{\times} \to \mathbb {S}^1$ be a unitary character. The local zeta integral is defined to be
$$ z(s,\omega,f) = \int_{F^{\times}} f(x)\omega(x)\omega_s(x)d^{\...
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Motivation and Interpretation of Classical and Geometric Satake Equivalence
I am struggling with understanding the motivation for both the classical and geometric forms of Satake equivalence. What prompted the development of this equivalence? Any open problems that it was ...
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Modularity in terms of the stack of elliptic curves
Modular forms can be viewed as something like global sections of (some tensor power of) the canonical line bundle on the stack $\mathcal M_{\text{ell}}$ of (generalized) elliptic curves. Is it ...
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Do all cuspidal automorphic representations of $\operatorname{GL}_2(\mathbb A_{\mathbb Q})$ come from Maass or holomorphic cusp forms?
A normalized cuspidal newform $f$ (either holomorphic or Maass) can be identified with a function on $\phi: \operatorname{GL}_2(\mathbb Q) \backslash \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, and ...
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Prerequisites for E. Frenkel's "Langlands correspondance for loop groups"
What would be the prerequisites to understand the material in this book ? It is advertised as an undergraduate course, but it seems this would apply to either very dedicated or quite advanced students ...
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How to recover abelian class field theory from Langlands?
I am reading about the Langlands program (mostly for fun). I am mostly self-taught in algebraic number theory. I have read that one recovers abelian class field theory from Langlands by setting $n=1$ ...
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An irreducible Weil(-Deligne) representation is automatically semisimple
Let $(V,r,N)$ be an irreducible Weil-Deligne representation. By irreducibility, $N=0$ automatically. Is it true that the Frobenius element acts semisimply?
This is not true if we only assume that the ...
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Quadratic reciprocity in Langlands program
I know quadratic reciprocity is the easiest example of langlands correspondence. Langlands correspondence gives some relation between automorphic forms and artin representations. My question is: what ...
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Satake correspondence for groups over finite field
In Langlands' program, Satake correspondence gives a correspondence between unramified representation of a reductive group $G$ over a local field and conjugacy classes in the Langlands dual group ${}^{...
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Langlands L functions for groups over finite fields.
In some reading on automorphic/Langlands-related papers I have seen some authors refer to the finite field analogues of Langlands objects, such as admissible representations, L factors but a simple ...
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Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
(1) Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know ...
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Descent for admissible representations of algebraic groups over local fields
Let $G$ be a reductive group and $F$ be a local field which is a finite extension of $\mathbb{Q}_p$. Assume $\Pi$ is an irreducible smooth admissible representation of $G(F)$ over $\bar{\mathbb{Q}}_l$,...
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Functional equation for $GL(3)\times GL(2)\times GL(1)$ L-functions
For two Maass forms
$$f(z)=\sum_{n\neq 0}a(n)\sqrt{2\pi y}K_{v_1-\frac{1}{2}}(2\pi|n|y)e^{2\pi inx}$$ and$$g(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m=1}^{\...
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Definition of Haar integral in Bushnell and Henniart
In Bushnell and Henniart's The Local Langland's Conjecture for GL(2) they define a right Haar integral on a locally profinite group $G$ as being a non-zero linear functional
$$
I: C^{\infty}_{c}(G) \...
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Results proved using perfectoid spaces understandable by an undergraduate
Many advanced areas of research in mathematics, the Riemann hypothesis, the Taniyama–Shimura conjecture, Green–Tao theorem etc. all have interesting consequences that could be stated using only ...
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Isomorphism of Weil representations only when $E/F$ is Galois?
$\DeclareMathOperator{\Ind}{Ind}\DeclareMathOperator{\Gal}{Gal}$Let $F$ be a $p$-adic field, and let $E \subseteq \overline{F}$ be a finite extension of $F$. Let $k_F$ and $k_E$ be the residue fields ...
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On the Fermat-Taylor-Wiles proof from integers to polynomials, and Langlands program
What is known as the Fermat's conjecture (or last theorem), proved by Taylor and Wiles (see The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles, 1995, Gerd Faltings, AMS notices or Fermat's ...
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Soft Question: in the Langlands program, which side is "geometric", which side is "spectral"?
I am in the process of ordering some book on the Langlands program, and learning more about it. In the mean time, I have a question which is easy to the experts, but being a beginner, I am not a 100% ...
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Does the Langlands program preserve CFT's distinction between local and global theories?
This question is vaguely related to: Different formulations of Class Field Theory
As I said there, I'm currently learning class field theory. For some motivation, I've also read a little about ...
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