Editing Poisson summation formula (section)

===Selberg trace formula===
{{main article|Selberg trace formula}}
Further generalization to [[locally compact abelian group]]s is required in [[number theory]]. In non-commutative [[harmonic analysis]], the idea is taken even further in the Selberg trace formula but takes on a much deeper character.

A series of mathematicians applying harmonic analysis to number theory, most notably Martin Eichler, [[Atle Selberg]], [[Robert Langlands]], and James Arthur, have generalised the Poisson summation formula to the Fourier transform on non-commutative locally compact reductive algebraic groups <math> G</math>  with a discrete subgroup <math> \Gamma</math>  such that <math> G/\Gamma</math>  has finite volume. For example, <math> G</math>  can be the real points of <math> SL_n</math>  and <math> \Gamma</math>  can be the integral points of <math> SL_n</math>. In this setting, <math> G</math>  plays the role of the real number line in the classical version of Poisson summation, and <math> \Gamma</math>  plays the role of the integers <math> n</math>  that appear in the sum. The generalised version of Poisson summation is called the Selberg Trace Formula and has played a role in proving many cases of Artin's conjecture and in Wiles's proof of Fermat's Last Theorem. The left-hand side of {{EquationNote|Eq.1}} becomes a sum over irreducible unitary representations of <math> G</math>, and is called "the spectral side," while the right-hand side becomes a sum over conjugacy classes of <math> \Gamma</math>, and is called "the geometric side."

The Poisson summation formula is the archetype for vast developments in harmonic analysis and number theory.