Davenport-Hasse relations and an explicit Langlands correspondence, II : twisting conjectures

Colin J. Bushnell; Guy Henniart

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 2, page 309-347
  • ISSN: 1246-7405

Abstract

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Let F / ℚ p be a finite field extension. The Langlands correspondence gives a canonical bijection between the set 𝒢 F 0 ( n ) of equivalence classes of irreducible n -dimensional representations of the Weil group 𝒲 F of F and the set 𝒜 F 0 ( n ) of equivalence classes of irreducible supercuspidal representations of GL n ( F ) . This paper is concerned with the case where n = p m . In earlier work, the authors constructed an explicit bijection π : 𝒢 F 0 ( p m ) → 𝒜 F 0 ( p m ) using a non-Galois tame base change map. If this tame base change satisfies a certain conjectured automorphic Davenport-Hasse relation, and there exists a Langlands correspondence in p -power degree, then π is the Langlands correspondence. This paper is concerned with the problem of showing, without assuming a priori the existence of the Langlands correspondence, that (on the Davenport-Hasse conjecture) π preserves local constants of pairs, and so is a Langlands correspondence. The principal obstruction is the lack of knowledge of certain elementary properties of the local constant ϵ ( π 1 × π 2 , s , ψ F ) for π i ∈ 𝒜 F 0 ( p m i ) . We state these properties as conjectures (which are certainly true, as consequences of the existence of the Langlands correspondence and analogous properties of the Langlands-Deligne local constant) and show that they imply the desired result: π is a Langlands correspondence. In the process, we prove several new unconditional results concerning π , and give a complete account of the rationality properties of L -functions and local constants of pairs for GL n ( F ) .

How to cite

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Bushnell, Colin J., and Henniart, Guy. "Davenport-Hasse relations and an explicit Langlands correspondence, II : twisting conjectures." Journal de théorie des nombres de Bordeaux 12.2 (2000): 309-347. <http://eudml.org/doc/248477>.

@article{Bushnell2000,
abstract = {Let $F/\mathbb \{Q\}_p$ be a finite field extension. The Langlands correspondence gives a canonical bijection between the set $\mathcal \{G\}^0_F (n)$ of equivalence classes of irreducible $n$-dimensional representations of the Weil group $\mathcal \{W\}_F$ of $F$ and the set $\mathcal \{A\}^0_F (n)$ of equivalence classes of irreducible supercuspidal representations of GL$_n(F)$. This paper is concerned with the case where $n = p^m$. In earlier work, the authors constructed an explicit bijection $\pi : \mathcal \{G\}^0_F (p^m) \rightarrow \mathcal \{A\}^0_F (p^m)$ using a non-Galois tame base change map. If this tame base change satisfies a certain conjectured automorphic Davenport-Hasse relation, and there exists a Langlands correspondence in $p$-power degree, then $\pi $ is the Langlands correspondence. This paper is concerned with the problem of showing, without assuming a priori the existence of the Langlands correspondence, that (on the Davenport-Hasse conjecture) $\pi $ preserves local constants of pairs, and so is a Langlands correspondence. The principal obstruction is the lack of knowledge of certain elementary properties of the local constant $\epsilon (\pi _1 \times \pi _2, s, \psi _F)$ for $\pi _i \in \mathcal \{A\}^0_F (p^\{m_i\})$. We state these properties as conjectures (which are certainly true, as consequences of the existence of the Langlands correspondence and analogous properties of the Langlands-Deligne local constant) and show that they imply the desired result: $\pi $ is a Langlands correspondence. In the process, we prove several new unconditional results concerning $\pi $, and give a complete account of the rationality properties of $L$-functions and local constants of pairs for GL$_n(F)$.},
author = {Bushnell, Colin J., Henniart, Guy},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {2},
pages = {309-347},
publisher = {Université Bordeaux I},
title = {Davenport-Hasse relations and an explicit Langlands correspondence, II : twisting conjectures},
url = {http://eudml.org/doc/248477},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Bushnell, Colin J.
AU - Henniart, Guy
TI - Davenport-Hasse relations and an explicit Langlands correspondence, II : twisting conjectures
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 2
SP - 309
EP - 347
AB - Let $F/\mathbb {Q}_p$ be a finite field extension. The Langlands correspondence gives a canonical bijection between the set $\mathcal {G}^0_F (n)$ of equivalence classes of irreducible $n$-dimensional representations of the Weil group $\mathcal {W}_F$ of $F$ and the set $\mathcal {A}^0_F (n)$ of equivalence classes of irreducible supercuspidal representations of GL$_n(F)$. This paper is concerned with the case where $n = p^m$. In earlier work, the authors constructed an explicit bijection $\pi : \mathcal {G}^0_F (p^m) \rightarrow \mathcal {A}^0_F (p^m)$ using a non-Galois tame base change map. If this tame base change satisfies a certain conjectured automorphic Davenport-Hasse relation, and there exists a Langlands correspondence in $p$-power degree, then $\pi $ is the Langlands correspondence. This paper is concerned with the problem of showing, without assuming a priori the existence of the Langlands correspondence, that (on the Davenport-Hasse conjecture) $\pi $ preserves local constants of pairs, and so is a Langlands correspondence. The principal obstruction is the lack of knowledge of certain elementary properties of the local constant $\epsilon (\pi _1 \times \pi _2, s, \psi _F)$ for $\pi _i \in \mathcal {A}^0_F (p^{m_i})$. We state these properties as conjectures (which are certainly true, as consequences of the existence of the Langlands correspondence and analogous properties of the Langlands-Deligne local constant) and show that they imply the desired result: $\pi $ is a Langlands correspondence. In the process, we prove several new unconditional results concerning $\pi $, and give a complete account of the rationality properties of $L$-functions and local constants of pairs for GL$_n(F)$.
LA - eng
UR - http://eudml.org/doc/248477
ER -

References

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