Jacquet–Langlands correspondence

In mathematics, the Jacquet–Langlands 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 Jacquet–Langlands correspondence relating automorphic representations of GL_r(D) and GL_dr(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 Jacquet–Langlands correspondence is bijection between

Automorphic representations of G of dimension greater than 1
Cuspidal automorphic representations of GL₂ that are square integrable (modulo the center) at each ramified place of G.

Corresponding representations have the same local components at all unramified places of G.

Rogawski (1983) and Deligne, Kazhdan & Vignéras (1984) extended the Jacquet–Langlands correspondence to division algebras of higher dimension.

References

Deligne, Pierre; Kazhdan, David; Vignéras, M.-F. (1984), "Représentations des algèbres centrales simples p-adiques", Représentations des groupes réductifs sur un corps local, Travaux en Cours, Paris: Hermann, pp. 33–117, 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., Zürich, pp. 1171–1182, 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): 161–196, doi:10.1215/s0012-7094-83-05006-8, ISSN 0012-7094, MR 0700135