Functoriality, Smith theory, and the Brauer homomorphism

Treumann, David; Venkatesh, Akshay

doi:10.4007/annals.2016.183.1.4

Cited by 10 publications

(37 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This conjecture is consistent with the general philosophy about linkage being compatible with the Langlands functorial transfer (Conjecture 6.3 in [27]).…”

Section: Overview Of the Papersupporting

confidence: 84%

“…Then r l pπq and r l pρq are in base change exactly when r l pπq plq -T 0 pρq. This conjecture is consistent with the general philosophy about linkage being compatible with the Langlands functorial transfer (Conjecture 6.3 in [27]).…”

Section: Overview Of the Papersupporting

confidence: 84%

“…The motivation to investigate cyclic base change in this situation comes from two conjectures of Treumann and Venkatesh (section 6.3 in [27]) where they investigate a particular case of Langlands functoriality, which we now recall.…”

Section: Motivationmentioning

confidence: 99%

“…Treumann and Venkatesh investigate Langlands functoriality in the following setup: let G be a reductive algebraic group defined over a p-adic field F and an automorphism σ P AutpGq of order l whose fixed point set is the algebraic group H " G σ . They introduce the notion of linkage 2 : a mod l representation π of G σ is linked to a σ-fixed mod l representation Π of G if the Frobenius twist π plq appears as a Jordan-Holder constituent of the Tate cohomology T ˚pΠq, which is naturally a G σ -representation (see section 3 and definition 6.2 in [27]). Under some technical assumptions, they conjecture that there exists an admissible Lhomomorphism L φ : L H ÝÑ L G such that if π is linked to the σ-invariant Π, then L φ sends the Langlands parameter of π into the one of Π.…”

Section: Motivationmentioning

confidence: 99%

See 3 more Smart Citations

Local base change via Tate cohomology

Ronchetti

2016

Represent. Theory

View full text Add to dashboard Cite

Abstract. We propose a new way to realize cyclic base change (a special case of Langlands functoriality) for prime degree extensions of characteristic zero local fields. Let F/E be a prime degree l extension of local fields of residue characteristic p = l. Let π be an irreducible cuspidal l-adic representation of GL n (E) and let ρ be an irreducible cuspidal l-adic representation of GL n (F ) which is Galois-invariant. Under some minor technical conditions on π and ρ (for instance, we assume that both are level zero) we prove that the mod lreductions r l (π) and r l (ρ) are in base change if and only if the Tate cohomology of ρ with respect to the Galois action is isomorphic, as a modular representation of GL n (E), to the Frobenius twist of r l (π). This proves a special case of a conjecture of Treumann and Venkatesh as they investigate the relationship between linkage and Langlands functoriality.

show abstract

“…This conjecture is consistent with the general philosophy about linkage being compatible with the Langlands functorial transfer (Conjecture 6.3 in [27]).…”

Section: Overview Of the Papersupporting

confidence: 84%

Section: Overview Of the Papersupporting

confidence: 84%

Section: Motivationmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

See 2 more Smart Citations

Local base change via Tate cohomology

Ronchetti

2016

Represent. Theory

View full text Add to dashboard Cite

show abstract

“…We choose q 1/2 ∈ F ℓ ; cf. [TV16,7. The reduction mod ℓ of H G,µ can be also regarded as a function…”

Section: Complements On Mod ℓ Representationsmentioning

confidence: 99%

Eichler-Shimura relations for local Shimura varieties

Koshikawa¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

Smith theory and cyclic base change functoriality

Feng

2024

Forum of Mathematics, Pi

View full text Add to dashboard Cite

Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbf {Z}/p\mathbf {Z}$ -extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For $\mathbf {Z}/p\mathbf {Z}$ -extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along $\mathbf {Z}/p\mathbf {Z}$ -extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from modular representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod p spherical Hecke algebras, in a joint appendix with Gus Lonergan.

show abstract

Functoriality, Smith theory, and the Brauer homomorphism

Abstract: If σ is an automorphism of order p of the semisimple group G, there is a natural correspondence between mod p cohomological automorphic forms on G and G σ . We describe this correspondence in the global and local settings.

Cited by 10 publications

References 41 publications

Local base change via Tate cohomology

Local base change via Tate cohomology

Eichler-Shimura relations for local Shimura varieties

Smith theory and cyclic base change functoriality

Contact Info

Product

Resources

About