Selberg trace formula
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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 operator]]s and [[differential operator]]s 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 formula (L-function)|explicit formulae]] of [[prime number]] theory. Here the closed [[geodesic]]s 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 operator]]s was linked to the '''Eichler-Selberg trace formula''', of Selberg and [[Martin Eichler]], for a Hecke operator acting on a vector space of [[cusp form]]s 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-function]]s associated to [[modular curve]]s; [[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 [[cusp]]s 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|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 (trace formula)|endoscopy]]). There is no definitive form of trace formula, in the sense that the ''L''<sup>2</sup> forms of index theorem have not actually caught up with all possible applications.
==Selberg trace formula for compact hyperbolic surfaces==
A compact hyperbolic surface <math>X</math> can be written as
:<math>\Gamma \backslash \mathbb{H}</math>,
where <math> \Gamma </math> is a subgroup of <math>PSL(2,\mathbb{R})</math>.
Then the spectrum for the [[Laplace-Beltrami operator]] on <math>X</math> is discrete (see [[discrete spectrum]]); that is
:<math> 0 = \mu_0 < \mu_1 \leq \mu_2 \leq \cdots </math>
where the eigenvalues <math> \mu_n </math> correspond to functions <math> u \in C^{\infty}(\mathbb{H}) </math>
such that
:<math>
\begin{cases}
u(\gamma z)=u(z), \ \ \forall \gamma \in \Gamma \\
y^2 \left (u_{xx} + u_{yy} \right) + \mu_{n} u = 0
\end{cases}
</math>.
Using the variable substitution
:<math> \mu = s(1-s), s=\frac{1}{2}+ir </math>
the eigenvalues are labeled
:<math> r_{n}, n \geq 0 </math>.
Then the '''Selberg trace formula''' is given by
:<math>
\sum_{n=0}^{\infty} h(r_n) = \left [ \frac{\mu(F)}{4 \pi } \int_{-\infty}^{\infty} r \, h(r) \tanh(\pi r) dr \right ] + \left [ \sum_{ \{T\} } \frac{ \log N(T_0) }{ N(T)^{1/2} - N(T)^{-1/2} } g \left ( \log N(T) \right ) \right ].
</math>
Here the sum <math> \lbrace T \rbrace </math> is taken over all distinct [[hyperbolic conjugacy class]]es,
the function <math> h </math> has to be an analytic function on <math> \vert \Im(r) \vert \leq 1/2+\delta </math>, satisfy
:<math> h(-r)=h(r), \ \vert h(r) \vert \leq M \left( 1+\vert \Re(r) \vert^{-2-\delta} \right ),</math>
where the numbers <math> \delta </math> and <math> M </math> are positive constants. The function <math> g </math> is the Fourier transform of <math> h </math>, that is,
<math> h(r) = \int_{-\infty}^{\infty} g(u) e^{iru} du </math>.
==External links==
*[http://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htm Selberg trace formula resource page]
==References==
* {{Citation | last1=Hejhal | first1=Dennis | title=The Selberg Trace Formula for PSL(2,R) | publisher=[[Springer-Verlag]] | location=Berlin, Heidelberg, New York | isbn=3-540-07988-2 | year=1976}}
[[Category:Automorphic forms]]
[[fr:Formule des traces de Selberg]]
[[zh:塞爾伯格跡公式]]