Jacquet–Langlands correspondence

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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) 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 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 Jacquet–Langlands correspondence is bijection between

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.