In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group.
The character is given by the trace of certain functions on G. The simplest case is when Γ is cocompact, when the representation breaks up into discrete summands.
When Γ is the cocompact subgroup Z of the real numbers G = R, the Selberg trace formula is essentially the Poisson summation formula.
The case when Γ\G is not compact is harder, because there is a continuous spectrum, described using Eisenstein series.
When Γ is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in terms of geometric data involving the lengths of geodesics on the Riemann surface.
In this case the Selberg trace formula is formally similar to the explicit formulas relating the zeros of the Riemann zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics.
Cases of particular interest include those for which the space is a compact Riemann surface S. The initial publication in 1956 of Atle Selberg dealt with this case, its Laplacian differential operator and its powers.
The traces of powers of a Laplacian can be used to define the Selberg zeta function.
Here the closed geodesics on S play the role of prime numbers.
At the same time, interest in the traces of Hecke operators was linked to the Eichler–Selberg trace formula, of Selberg and Martin Eichler, for a Hecke operator acting on a vector space of cusp forms of a given weight, for a given congruence subgroup of the modular group.
Here the trace of the identity operator is the dimension of the vector space, i.e. the dimension of the space of modular forms of a given type: a quantity traditionally calculated by means of the Riemann–Roch theorem.
The trace formula has applications to arithmetic geometry and number theory.
For instance, using the trace theorem, Eichler and Shimura calculated the Hasse–Weil L-functions associated to modular curves; Goro Shimura's methods by-passed the analysis involved in the trace formula.
The development of parabolic cohomology (from Eichler cohomology) provided a purely algebraic setting based on group cohomology, taking account of the cusps characteristic of non-compact Riemann surfaces and modular curves.
The trace formula also has purely differential-geometric applications.
For instance, by a result of Buser, the length spectrum of a Riemann surface is an isospectral invariant, essentially by the trace formula.
A compact hyperbolic surface X can be written as the space of orbits
The Selberg trace formula for this case is easier than the general case because the surface is compact so there is no continuous spectrum, and the group Γ has no parabolic or elliptic elements (other than the identity).
where the eigenvalues μn correspond to Γ-invariant eigenfunctions u in C∞(H) of the Laplacian; in other words
The right hand side is a sum over conjugacy classes of the group Γ, with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes {T } (which are all hyperbolic in this case).
a compactly supported continuous function on G. The trace formula in this setting is the following equality:
Define the following operator on compactly supported functions on
denote a collection of representatives of conjugacy classes in
is a semisimple Lie group with a maximal compact subgroup
can be described in geometric terms using the compact Riemannian manifold (more generally orbifold)
The orbital integrals and the traces in irreducible summands can then be computed further and in particular one can recover the case of the trace formula for hyperbolic surfaces in this way.
The general theory of Eisenstein series was largely motivated by the requirement to separate out the continuous spectrum, which is characteristic of the non-compact case.
The case of SL2(C) is discussed in Gel'fand, Graev & Pyatetskii-Shapiro (1990) and Elstrodt, Grunewald & Mennicke (1998).
Gel'fand et al also treat SL2(F) where F is a locally compact topological field with ultrametric norm, so a finite extension of the p-adic numbers Qp or of the formal Laurent series Fq((T)); they also handle the adelic case in characteristic 0, combining all completions R and Qp of the rational numbers Q.
Contemporary successors of the theory are the Arthur–Selberg trace formula applying to the case of general semisimple G, and the many studies of the trace formula in the Langlands philosophy (dealing with technical issues such as endoscopy).