From Wikipedia, the free encyclopedia
In mathematics, the JacquetâLanglands correspondence is a correspondence between
automorphic forms on
GL2 and its twisted forms, proved by
Jacquet and
Langlands (
1970, section 16) in their book
Automorphic Forms on GL(2) using the
Selberg trace formula. It was one of the first examples of the
Langlands philosophy that maps between
L-groups should induce maps between
automorphic representations. There are generalized versions of the JacquetâLanglands correspondence relating automorphic representations of GLr(D) and GLdr(F), where D is a
division algebra of degree d2 over the
local or
global field F.
Suppose that G is an inner twist of the algebraic group GL2, in other words the
multiplicative group of a
quaternion algebra. The JacquetâLanglands correspondence is
bijection between
Corresponding representations have the same local components at all unramified places of G.
Rogawski (1983) and
Deligne, Kazhdan & VignĂ©ras (1984) extended the JacquetâLanglands correspondence to division algebras of higher dimension.
References
-
Deligne, Pierre;
Kazhdan, David;
Vignéras, M.-F. (1984), "Représentations des algÚbres centrales simples p-adiques",
ReprĂ©sentations des groupes rĂ©ductifs sur un corps local, Travaux en Cours, Paris: Hermann, pp. 33â117,
ISBN
978-2-7056-5989-9,
MR
0771672
- Henniart, Guy (2006), "On the local Langlands and Jacquet-Langlands correspondences", in
Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; et al. (eds.),
International Congress of Mathematicians. Vol. II, Eur. Math. Soc., ZĂŒrich, pp. 1171â1182,
ISBN
978-3-03719-022-7,
MR
2275640, archived from
the original on 2012-03-15, retrieved 2011-07-01
-
Jacquet, H.;
Langlands, Robert P. (1970),
Automorphic Forms on GL(2), Lecture Notes in Mathematics, vol. 114, Berlin, New York:
Springer-Verlag,
doi:
10.1007/BFb0058988,
ISBN
978-3-540-04903-6,
MR
0401654
- Rogawski, Jonathan D. (1983),
"Representations of GL(n) and division algebras over a p-adic field",
Duke Mathematical Journal, 50 (1): 161â196,
doi:
10.1215/s0012-7094-83-05006-8,
ISSN
0012-7094,
MR
0700135