Selberg trace formula

In mathematics, the Selberg trace formula is a central result, or area of research, in non-commutative harmonic analysis. It provides an expression for the trace, in a sense suitably generalising that of the trace of a matrix, for suitable integral operators and differential operators acting in spaces of functions defined on a homogeneous space G/Γ where G is a Lie group and Γ a discrete group, or more generally a double coset space H\G/Γ.

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, in a case such as this, provide a kind of zeta function (see Selberg zeta function). The immediate interest of this case was the powerful 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. The relationship was immediately recognised as a significant commentary on the Riemann hypothesis. The trace formula was singled out as the non-commutative generalisation of the Poisson summation formula.

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 would be 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. This development made it clear that further information was available, by methods (which would come to be seen as naturally described as those) of representation theory.

Development

A large number of developments followed. The Eichler-Shimura theorem 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. In the end the compact quotient case of the Selberg trace formula was more-or-less absorbed into the theory of the Atiyah-Singer index theorem; but the non-compact case is met immediately when Γ is taken to be an arithmetic group.

Later work

In the 1960s the general thrust of the Selberg trace formula, as a piece of analysis, was taken up by the Israel Gelfand school, by Harish-Chandra and Langlands in Princeton, and by Tomio Kubota in Japan. 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 existence of trace formulae both for the differential operator and Hecke operator cases was a hint of the power (for essentially arithmetic cases) of the adele group approach.

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). There is no definitive form of trace formula, in the sense that the L² forms of index theorem have not actually caught up with all possible applications.

Selberg trace formula for compact hyperbolic surfaces

A compact hyperbolic surface X can be written as

$\Gamma \backslash \mathbb {H}$,

where $\Gamma$ is a subgroup of $PSL(2,\mathbb {R} )$.

Then the spectrum for the Laplace-Beltrami operator on X is discrete (see discrete spectrum); that is

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

where the eigenvalues $\mu _{n}$ correspond to functions $u\in C^{\infty }(\mathbb {H} )$ such that

$y^{2}(u_{xx}+u_{yy})+\mu _{n}u=0$.

and that satisfy

$u(\gamma z)=u(z)$

for all elements $\gamma \in \Gamma$.

Using the variable substitution

$\mu =s(1-s),s={\frac {1}{2}}+ir$

the eigenvalues are labeled

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

Then the Selberg trace formula is given by

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

Here the sum $\{T\}$ is taken over all distinct hyperbolic conjugacy classes, the function h has to be an analytic function on $\vert \Im (r)\vert \leq 1/2+\delta$, satisfy $h(-r)=h(r)$ and $\vert h(r)\vert \leq M(1+\vert \Re (r)\vert ^{-2-\delta }$ where the numbers $\delta$ and M are positive constants. The function g is the Fourier transform of h, that is, $h(r)=\int _{-\infty }^{\infty }g(u)e^{iru}du$.

External link

• Selberg trace formula resource page