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. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. 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. Selberg worked out the non-compact case when G is the group SL(2, R); the extension to higher rank groups is the Arthur–Selberg trace formula.

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. Motivated by the analogy, Selberg introduced the Selberg zeta function of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula.

Early history

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. The interest of this case was the analogy between the formula obtained, and the explicit formulae of prime number theory. 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.

Applications

The trace formula has applications to arithmetic geometry^{[citation needed]} 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.

Later work

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.^[1]

The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroup Γ into an algebraic group over a field which is technically easier to work with. 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 Q_p or of the formal Laurent series F_q((T)); they also handle the adelic case in characteristic 0, combining all completions R and Q_p 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). The Selberg trace formula can be derived from the Arthur–Selberg trace formula with some effort.

Selberg trace formula for compact hyperbolic surfaces

A compact hyperbolic surface X can be written as the space of orbits

$${\displaystyle \Gamma \backslash \mathbf {H} ,}$$

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).

Then the spectrum for the Laplace–Beltrami operator on X is discrete and real, since the Laplace operator is self adjoint with compact resolvent; that is

$${\displaystyle 0=\mu _{0}<\mu _{1}\leq \mu _{2}\leq \cdots }$$

$${\displaystyle {\begin{cases}u(\gamma z)=u(z),\qquad \forall \gamma \in \Gamma \\y^{2}\left(u_{xx}+u_{yy}\right)+\mu _{n}u=0.\end{cases))}$$

Using the variable substitution

$${\displaystyle \mu =s(1-s),\qquad s={\tfrac {1}{2))+ir}$$

$${\displaystyle r_{n},n\geq 0.}$$

Then the Selberg trace formula is given by

$${\displaystyle \sum _{n=0}^{\infty }h(r_{n})={\frac {\mu (X)}{4\pi ))\int _{-\infty }^{\infty }r\,h(r)\tanh(\pi r)\,dr+\sum _{\{T\)){\frac {\log N(T_{0})}{N(T)^{\frac {1}{2))-N(T)^{-{\frac {1}{2))))}g(\log N(T)).}$$

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). The function h has to satisfy the following:

• be analytic on |Im(r)| ≤ 1/2 + δ;
• h(−r) = h(r);
• there exist positive constants δ and M such that:
  $${\displaystyle \vert h(r)\vert \leq M\left(1+\left|\operatorname {Re} (r)\right|\right)^{-2-\delta }.}$$

  

The function g is the Fourier transform of h, that is,

$${\displaystyle h(r)=\int _{-\infty }^{\infty }g(u)e^{iru}\,du.}$$