In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003.
It describes the character of the representation of G(A) on the discrete part L20(G(F)\G(A)) of L2(G(F)\G(A)) in terms of geometric data, where G is a reductive algebraic group defined over a global field F and A is the ring of adeles of F. There are several different versions of the trace formula.
The first version was the unrefined trace formula, whose terms depend on truncation operators and have the disadvantage that they are not invariant.
The simple trace formula (Flicker & Kazhdan 1988) is less general but easier to prove.
Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups.
The group G acts on the space of functions on Γ\G by the right regular representation R, and this extends to an action of the group ring of G, considered as the ring of functions f on G. The character of this representation is given by a generalization of the Frobenius formula as follows.
The original Selberg trace formula studied a discrete subgroup Γ of a real Lie group G(R) (usually SL2(R)).
The version of the trace formula above is not particularly easy to use in practice, one of the problems being that the terms in it are not invariant under conjugation.
Two elements of a group G(F) are called stably conjugate if they are conjugate over the algebraic closure of the field F. The point is that when one compares elements in two different groups, related for example by inner twisting, one does not usually get a good correspondence between conjugacy classes, but only between stable conjugacy classes.
For example, if the functions f are cuspidal, which means that for any unipotent radical N of a proper parabolic subgroup (defined over F) and any x, y in G(A), then the operator R(f) has image in the space of cusp forms so is compact.
The Arthur–Selberg trace formula can be used to study similar correspondences on higher rank groups.
It can also be used to prove several other special cases of Langlands functoriality, such as base change, for some groups.
Kottwitz (1988) used the Arthur–Selberg trace formula to prove the Weil conjecture on Tamagawa numbers.
Lafforgue (2002) described how the trace formula is used in his proof of the Langlands conjecture for general linear groups over function fields.