The local Langlands conjectures are certain conjectures in the context of the Langlands program. Where the genuine Langlands correspondence concerns global fields, the local Langlands correspondence concerns local fields.
In the case of the algebraic group (the general linear group), the local Langlands conjectures assert a correspondence between:
F-semisimple Weil-Deligne representations (see Def. below) of the Weil group of a local field ;
irreducible admissible representations of (see Def. below),
generalizing local class field theory from abelian Galois groups to non-abelian Galois groups.
More generally, it states that for a local field and a reductive group , the isomorphism classes of irreducible admissible representations of are partitioned into L-packets by the equivalence classes of L-parameters.
We discuss some of the basic definitions, see also Tate 77.
Throughout:
is a p-adic field with denoting its residue field;
is a reductive group.
(irreducible and admissible representations) A linear representation of on a vector space is called:
irreducible if the only -invariant subspaces are and .
admissible if for any open subgroup of the fixed subspace is finite-dimensional.
(Weil group)
The Weil group is the subgroup of the Galois group defined as the inverse image of Frobenius automorphisms under the surjective map .
(Weil-Deligne representation)
A Weil-Deligne representation is a pair where is a representation of the Weil group (Def. ) and is a nilpotent monodromy operator satisfying for all , where is the valuation of the corresponding element of under the isomorphism of local class field theory.
Let be a -adic field. The only irreducible admissible representations of () are continuous group homomorphisms . The -dimensional Weil-Deligne representations of must have monodromy operator and must factor through the abelianization . The local Langlands correspondence in this case is the same as local class field theory, and sends to the Weil-Deligne representation , where is the composition , with being the Artin reciprocity map of local class field theory.
Let be a -adic field, . Given continuous admissible characters and , we define to be the vector space of function which are locally constant with respect to the -adic topology on and satisfying
Letting act on this by translation gives us a representation of . This representation is said to belong to the principal series.
Whether a principal series is irreducible or not is governed by the following theorem:
[Bernstein-Zelevinsky]
If then is irreducible.
If then we have an exact sequence
where is the -dimensional representation of given by , and is an irreducible representation.
If then we have an exact sequence
with and as above.
Therefore the irreducible admissible representations of are the following:
Principal series representations for .
Special representations .
-dimensional representations .
“Base change” representations for a quadratic extension of and an admissible character .
Let be the representation of the Weil group associated to the character by the local Langlands correspondence for . Then the local Langlands correspondence associates to each irreducible admissible representation of a -dimensional Weil-Deligne representation as follows:
To the principal series representation we associate the Weil-Deligne representation .
To the special representation , we associate the Weil-Deligne representation .
To the -dimensional representation , we associate the Weil-Deligne representation .
To the “base change” representation we associate the Weil-Deligne representation (), where is the unique nontrivial element of .
Laurent Fargues and Peter Scholze have developed a geometric approach to the local Langlands conjecture in FarguesScholze21. In this formulation the local Langlands correspondence is expressed as an equivalence between the derived category of certain -adic sheaves (this requires the theory of solid abelian groups? from condensed mathematics) on the moduli stack of -torsors on the Fargues-Fontaine curve on one hand, and the derived category of certain coherent sheaves on the moduli stack of L-parameters (denoted in FarguesScholze21) in the other (compare geometric Langlands correspondence). More precisely:
There is an equivalence
of stable -categories equipped with actions of .
In their work Fargues and Scholze are able to associate an L-parameter to a Schur object in using the method of Vincent Lafforgue (originally developed for the global Langlands correspondence for function fields over ) involving excursion operators. This is also known as the “automorphic to Galois” direction in the global setting.
In the other direction (analogous to the “Galois to automorphic direction”) Fargues and Scholze approach this via the construction of the spectral action, which involves the use of infinity groupoids? (which they refer to as anima). Subsequent work of Johannes Anschütz and Arthur-Cesar Le Bras in AnschutzLeBras21 use this spectral action to show the existence of the Hecke eigensheaf.
The p-adic local Langlands correspondence concerns -adic representations of the absolute Galois group of a -adic field , as opposed to complex Weil-Deligne or -adic representations. It is related to p-adic Hodge theory. It should also be compatible with a mod p local Langlands correspondence, which has coefficients in finite fields. Currently the only known case of the -adic and mod local Langlands correspondences is the case , see also Breuil2010.
See also:
Wikipedia, Local Langlands conjectures.
Colin Bushnell, Guy Henniart, The local Langlands conjecture for
Topics on Automorphic Forms, course notes by Chao Li from a course taught by Jack Thorne
Basic definitions are discussed in:
An approach via the Fargues-Fontaine curve:
The above work is partially inspired by the following work on the global Langlands correspondence for function fields over :
globale_ arXiv:1209.5352
The spectral action of Fargues and Scholze is used to show the existence of the Hecke eigensheaf in
The -adic and mod local Langlands correspondence is discussed in:
Last revised on July 23, 2023 at 17:38:07. See the history of this page for a list of all contributions to it.