Jacquet–Langlands correspondence

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

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

s on GL₂ and its twisted forms, proved by in their book ''

Automorphic Forms on GL(2) ''Automorphic Forms on GL(2)'' is a mathematics book by where they rewrite Erich Hecke's theory of modular forms in terms of the representation theory of GL(2) over local fields and adele rings of global field In mathematics, a global field is one ...

'' using the

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

. It was one of the first examples of the Langlands philosophy that maps between L-groups should induce maps between

automorphic representations In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

. 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 In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a f ...

of degree ''d''² over the

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