Questions tagged [langlands-conjectures]
Higher reciprocity laws
413 questions
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Compatibility of Langlands classification and Bushnell-Kutzko types
Bushnell-Kutzko construct for $G=\mathrm{GL}_n(F)$ equivalences between any Bernstein block of $G$ and the category of modules for a tensor product of affine Hecke algebras $\bigotimes_i H_{m_i}(q^{...
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Have there been major developments in the Langlands Program in the number field setting since 2010?
By the Langlands Program, I primarily mean the existence of a natural bijection, satisfying the desired properties, between algebraic cuspidal automorphic forms for $GL_n$ over a number field $K$, and ...
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Do odd-weight Galois representations have positive and negative eigenspaces for the infinite Frobenius?
Suppose $V$ is a geometric $\ell$-adic Galois representation of $G_{\mathbb{Q}} = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, pure of weight $2n+1$ for $n \in \mathbb{Z}$. This means that $V$ is ...
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Why did we miss a cohomology theory of “Weil type”?
It’s a well known fact due to Serre that there is no $\mathbf Q_p$- and $\mathbf R$-valued cohomology theory of “Weil type” for varieties over $\overline{\mathbf F_p}$; but for $\ell\ne p$, a $\mathbf ...
- 1,186
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What is the inertia-Deligne group?
In Peter Scholze’s “Geometrization of Local Langlands, Motivically”, he discusses an object coined as the “inertia-Deligne group”, which is an object that lies in an extension of the inertia group by ...
- 75
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Relationship between Shimura varieties and the moduli stack of principal bundles on a curve
I'm aware of two "geometric" constructions of automorphic forms. The first is in terms of Shimura varieties. A Shimura variety is (roughly speaking) a moduli space of abelian varieties with ...
- 2,275
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The dual group of a G-variety
In the paper https://arxiv.org/abs/1702.08264 proposition 5.4, a weak spherical datum for a $G$-variety $X$ is defined, I listened a talk yesterday https://mathematics.jhu.edu/event/number-theory-...
- 411
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Fourier transform on basic affine space as a normalized intertwining operator
There is a Fourier transform $F_{\omega}$ acting on the basic affine space defined in the paper by Braverman and Kazhdan On the Schwartz space of the basic affine space as a generalization of the ...
- 411
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221
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How could we get the Weil group for global function fields?
Thanks for your help. I know the definition of Weil group and how to get it for $p$-adic local field cases. By reading the answer https://math.stackexchange.com/a/2898449/1007843, I write down my ...
- 473
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Point-wise spectral actions in de Rham geometric Langlands
Let $(G, G^{\vee})$ be a pair of Langlands dual groups. Let $Bun_G$ and $LocSys_{G^{\vee}}$ denote the moduli stacks of $G$-bundles and $G^{\vee}$ local systems over a fixed smooth projective curve.
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Morphisms in S-points of Hecke stacks
Let $X$ be a smooth projective curve over $\mathbb C$, $G$ be a connected reductive group over $\mathbb C$. The Hecke stacks is usually defined as the stacks whose $S$-valued points (for any $\mathbb ...
- 405
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Hecke compatibility for p-adic local langlands
Both in the global geometric Langlands as well as the local ($l \neq p$) Langlands formulated in the paper Geometrization of the local Langlands correspondence by Fargues and Scholze, there is a ...
- 131
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Can two irreps of a p-adic group with distinct cuspidal support have a non-trivial extension between them?
Let $G$ be a connected reductive algebraic group over a $p$-adic field $F$. All representations mentioned here are assumed to be smooth. Recall that a cuspidal pair $(M, \sigma)$ is a Levi subgroup $...
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1
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259
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Automorphic representations with given $\pi_p$ ($G=\operatorname{GL}_2$ and $\operatorname{GSp}_4$)
Let $G$ be a reductive group over $\mathbb Q$ such that $G(\mathbb R)$ admits a holomorphic discrete series $\pi_\infty$. Maybe well-known to experts doing analytic number theory (or vertical Sato–...
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When does a Langlands parameter really occur in cohomology of local Shimura varieties?
Let $G$ be a connected reductive group over $\mathbb Q_p$. Let $\{\mu \}$ be a conjugacy class of cocharacters $\mu: \mathbb G_m \to G$ over $\mathbb Q_p^{alg}$, and $\{ b \} \in G(\breve{\mathbb Q}_p)...
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