1. References

In mathematics, the Jacquet–Langlands correspondence is a correspondence between automorphic forms on GL₂ and its twisted forms, proved by {{Harvs|txt|author1-link=Hervé Jacquet|last=Jacquet|author2-link=Robert Langlands|last2=Langlands|year=1970|loc=section 16}} 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.

References

• {{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Kazhdan | first2=David | author2-link=David Kazhdan | last3=Vignéras | first3=M.-F. | author3-link=Marie-France Vign%C3%A9ras | title=Représentations des groupes réductifs sur un corps local | url=https://books.google.com/books?id=tQHvAAAAMAAJ | publisher=Hermann | location=Paris | series=Travaux en Cours | isbn=978-2-7056-5989-9 | mr=771672 | year=1984 | chapter=Représentations des algèbres centrales simples p-adiques | pages=33–117}}
• {{Citation | last1=Henniart | first1=Guy | editor1-last=Sanz-Solé | editor1-first=Marta | editor1-link=Marta Sanz-Solé | editor2-last=Soria | editor2-first=Javier | editor3-last=Varona | editor3-first=Juan Luis |display-editors = 3 | editor4-last=Verdera | editor4-first=Joan | title=International Congress of Mathematicians. Vol. II | url=http://mathunion.org/ICM/ICM2006.2/ | publisher=Eur. Math. Soc., Zürich | isbn=978-3-03719-022-7 | mr=2275640 | year=2006 | chapter=On the local Langlands and Jacquet-Langlands correspondences | pages=1171–1182}}
• {{Citation | author2-link=Robert Langlands | last1=Jacquet | first1=H. | last2=Langlands | first2=Robert P. | title=Automorphic forms on GL(2) | url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0058988 | mr=0401654 | year=1970 | volume=114 | isbn=978-3-540-04903-6}}
• {{Citation | last1=Rogawski | first1=Jonathan D. | title=Representations of GL(n) and division algebras over a p-adic field | url=http://projecteuclid.org/getRecord?id=euclid.dmj/1077303004 | mr=700135 | year=1983 | journal=Duke Mathematical Journal | issn=0012-7094 | volume=50 | issue=1 | pages=161–196 | doi=10.1215/s0012-7094-83-05006-8}}

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