In mathematics, the Jacquet–Langlands correspondence is a correspondence between automorphic form

Automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

s on GL₂ and its twisted forms, proved by using the Selberg trace formula

Selberg trace formula

In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of G on the space L2 of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group...

. It was one of the first examples of the Langlands philosophy that maps between L-group

L-group

In mathematics, L-group may refer to the following groups:* The Langlands dual, LG, of a reductive algebraic group G* A group in L-theory, L...

s 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

Quaternion algebra

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars , i.e...

. 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.

and extended the Jacquet–Langlands correspondence to division algebras of higher dimension.