Editing Selberg trace formula

{{Short description|Mathematical theorem}}
In [[mathematics]], the '''Selberg trace formula''', introduced by  {{harvtxt|Selberg|1956}}, is an expression for the character of the [[unitary representation]] of a [[Lie group]] {{mvar|G}} on the space {{math|''L''<sup>2</sup>(Γ\''G'')}} of [[square-integrable function]]s, where {{math|Γ}} is a cofinite [[discrete group]]. The character is given by the trace of certain functions on {{mvar|G}}.

The simplest case is when {{math|Γ}} is [[cocompact group action|cocompact]], when the representation breaks up into discrete summands. Here the trace formula is an extension of the [[Induced representation#Alternate formulations|Frobenius formula]] for the character of an [[induced representation]] of finite groups. When {{math|Γ}} is the cocompact subgroup {{math|'''Z'''}} of the real numbers {{math|''G'' {{=}} '''R'''}}, the Selberg trace formula is essentially the [[Poisson summation formula]].

The case when {{math|Γ\''G''}} is not compact is harder, because there is a [[spectrum (functional analysis)|continuous spectrum]], described using [[Eisenstein series]]. Selberg worked out the non-compact case when {{mvar|G}} is the group {{math|SL(2, '''R''')}}; the extension to higher rank groups is the [[Arthur–Selberg trace formula]].

When {{math|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 formulae (L-function)|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]] {{mvar|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 formula (L-function)|explicit formulae]] of [[prime number]] theory. Here the [[closed geodesic]]s on {{mvar|S}} play the role of prime numbers.

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 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]] and [[number theory]].  For instance, using the trace theorem, [[Eichler–Shimura congruence relation|Eichler and Shimura]] 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 form|cusps]] characteristic of non-compact Riemann surfaces and modular curves.

The trace formula also has purely [[differential geometry|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.

==Selberg trace formula for compact hyperbolic surfaces==
A compact hyperbolic surface {{mvar|X}} can be written as the space of orbits <math display="block">\Gamma \backslash \mathbf{H},</math> where {{math|Γ}} is a subgroup of {{math|PSL(2, '''R''')}}, and {{math|'''H'''}} is the [[upper half plane]], and {{math|Γ}} acts on {{math|'''H'''}} by [[linear fractional transformations]].

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 {{math|Γ}} has no parabolic or elliptic elements (other than the identity).

Then the spectrum for the [[Laplace–Beltrami operator]] on {{mvar|X}} is discrete and real, since the Laplace operator is self adjoint with compact [[resolvent formalism|resolvent]]; that is
<math display="block"> 0 = \mu_0 < \mu_1 \leq \mu_2 \leq \cdots </math>
where the eigenvalues {{math|''μ<sub>n</sub>''}} correspond to {{math|Γ}}-invariant eigenfunctions {{mvar|u}} in {{math|''C''<sup>∞</sup>('''H''')}} of the Laplacian; in other words
<math display="block">\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}</math>

Using the variable substitution
<math display="block"> \mu = s(1-s), \qquad  s=\tfrac{1}{2}+ir </math>
the eigenvalues are labeled
<math display="block"> r_{n}, n \geq 0. </math>

Then the '''Selberg trace formula''' is given by
<math display="block">\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)). </math>

The right hand side is a sum over conjugacy classes of the group {{math|Γ}}, with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes {{math|{''T''&thinsp;} }} (which are all hyperbolic in this case). The function {{mvar|h}} has to satisfy the following:

* be analytic on {{math|{{!}}Im(''r''){{!}} ≤ {{sfrac|1|2}} + ''δ''}}; 
* {{math|''h''(−''r'') {{=}} ''h''(''r'')}};
* there exist positive constants {{mvar|δ}} and {{mvar|M}} such that: <math display="block">\vert h(r) \vert \leq M \left( 1+\left| \operatorname{Re}(r) \right| \right )^{-2-\delta}.</math>

The function {{mvar|g}} is the Fourier transform of {{mvar|h}}, that is,
<math display="block"> h(r) = \int_{-\infty}^\infty g(u) e^{iru} \, du. </math>

==The general Selberg trace formula for cocompact quotients==
===General statement===
Let ''G'' be a unimodular locally compact group, and <math>\Gamma</math> a discrete cocompact subgroup of ''G'' and <math>\phi</math> a compactly supported continuous function on ''G''. The trace formula in this setting is the following equality: 
<math display=block>\sum_{\gamma\in\{\Gamma\}} a_\Gamma^G(\gamma)\int_{G^\gamma\setminus G}\phi(x^{-1}\gamma x)\,dx = \sum_{\pi\in\widehat G}a_\Gamma^G(\pi)\operatorname{tr}\pi(\phi)</math>
where <math>\{\Gamma\}</math> is the set of conjugacy classes in <math>\Gamma</math>, <math>\widehat G</math> is the [[unitary dual]] of ''G'' and:
* for an element <math>\gamma \in \Gamma</math>, <math> a_\Gamma^G(\gamma) = \text{volume}(\Gamma^\gamma\setminus G^\gamma).</math> with <math>G_\gamma, \Gamma_\gamma</math> the centralisers of <math>\gamma</math> in <math>G,\Gamma</math> respectively; 
* for an [[irreducible representation|irreducible]] [[unitary representation]] <math>\pi</math> of <math>G</math>, <math>a_\Gamma^G(\pi)</math> is the [[Unitary_representation#Complete_reducibility|multiplicity]] of <math>\pi</math> in the right-representation on <math>\Gamma\backslash G</math> in <math>L^2(\Gamma\backslash G</math>), and <math>\pi(\phi)</math> is the operator <math>\int_G \phi(g)\pi(g) dg</math>; 
* all integrals and volumes are taken with respect to the [[Haar measure]] on <math>G</math> or its quotients. 

The left-hand side of the formula is called the ''geometric side'' and the right-hand side the ''spectral side''. The terms <math>\int_{G^\gamma\setminus G}\phi(x^{-1}\gamma x)\,dx</math> are [[orbital integral]]s. 

===Proof===
Define the following operator on compactly supported functions on <math>\Gamma\backslash G</math>: 
<math display=block>R(\phi) = \int_G \phi(x)R(x)\,dx,</math>
It extends continuously to <math>L^2(\Gamma\setminus G)</math> and for <math>f\in L^2(\Gamma\setminus G)</math> we have:
<math display=block>(R(\phi)f)(x) = \int_G\phi(y)f(xy)\,dy = \int_{\Gamma\setminus G}\left(\sum_{\gamma\in\Gamma}\phi(x^{-1}\gamma y)\right)f(y)\,dy</math>
after a change of variables. Assuming <math>\Gamma\setminus G</math> is compact, the operator <math>R(\phi)</math> is [[trace-class]] and the trace formula is the result of computing its trace in two ways as explained below.<ref>This presentation is from {{cite book|author=Arthur|chapter=The trace formula and Hecke operators|title=Number theory, trace formulas and discrete groups|publisher=Academic Press|year=1989}}</ref>

The trace of <math>R(\phi)</math>can be expressed as the integral of the kernel <math>K(x,y)=\sum_{\gamma\in\Gamma}\phi(x^{-1}\gamma y)</math> along the diagonal, that is:
<math display=block>\operatorname{tr}R(\phi) = \int_{\Gamma\setminus G}\sum_{\gamma\in\Gamma}\phi(x^{-1}\gamma x)\,dx.</math>
Let <math>\{\Gamma\}</math> denote a collection of representatives of conjugacy classes in <math>\Gamma</math>, and <math>\Gamma^\gamma</math> and <math>G^\gamma</math> the respective centralizers of <math>\gamma</math>.  
Then the above integral can, after manipulation, be written
<math display=block>\operatorname{tr}R(\phi) = \sum_{\gamma\in\{\Gamma\}} a_\Gamma^G(\gamma)\int_{G^\gamma\setminus G}\phi(x^{-1}\gamma x)\,dx.</math>
This gives the ''geometric side'' of the trace formula.

The ''spectral side'' of the trace formula comes from computing the trace of <math>R(\phi)</math> using the decomposition of the regular representation of <math>G</math> into its irreducible components. Thus
<math display=block>\operatorname{tr}R(\phi) = \sum_{\pi\in\hat G}a_\Gamma^G(\pi)\operatorname{tr}\pi(\phi)</math>
where <math>\hat G</math> is the set of irreducible unitary representations of <math>G</math> (recall that the positive integer <math>a_\Gamma^G(\pi)</math> is the multiplicity of <math>\pi</math> in the unitary representation <math>R</math> on <math>L^2(\Gamma\setminus G)</math>). 

===The case of semisimple Lie groups and symmetric spaces===
When <math>G</math> is a semisimple Lie group with a maximal compact subgroup <math>K</math> and <math>X=G/K</math> is the associated [[symmetric space]] the conjugacy classes in <math>\Gamma</math> can be described in geometric terms using the compact Riemannian manifold (more generally orbifold) <math>\Gamma \backslash X</math>. 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.

==Later work==
The general theory of [[Eisenstein series]] was largely motivated by the requirement to separate out the [[spectrum (functional analysis)|continuous spectrum]], which is characteristic of the non-compact case.<ref>{{harvnb|Lax|Phillips|1980}}</ref>

The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroup {{math|Γ}} into an algebraic group over a field which is technically easier to work with. The case of SL<sub>2</sub>('''C''') is discussed in {{harvtxt|Gel'fand|Graev|Pyatetskii-Shapiro|1990}} and {{harvtxt|Elstrodt|Grunewald|Mennicke|1998}}. Gel'fand et al also treat SL<sub>2</sub>({{mvar|''F''}}) where {{mvar|F}} is a locally compact topological field with [[ultrametric space|ultrametric norm]], so a finite extension of the [[p-adic numbers]] '''Q'''<sub>''p''</sub> or of the [[formal Laurent series]] '''F'''<sub>''q''</sub>((''T'')); they also handle the adelic case in characteristic 0, combining all completions '''R''' and '''Q'''<sub>''p''</sub> 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 (trace formula)|endoscopy]]). The Selberg trace formula can be derived from the Arthur–Selberg trace formula with some effort.

== See also ==

* [[Jacquet–Langlands correspondence]]

== Notes ==
{{Reflist}}
== References ==
* {{cite book| mr =1482800 | last = Borel | first = Armand | author-link = Armand Borel | title = 
Automorphic forms on SL<sub>2</sub>(R) | series = Cambridge Tracts in Mathematics | volume = 130
| publisher = [[Cambridge University Press]] | year = 1997 | isbn = 0-521-58049-8 | url = https://www.cambridge.org/core/books/automorphic-forms-on-sl2-r/20CDEAD276B61040B2B9013A21ACC145 }}
* {{cite book|mr = 0768584 | last1 = Chavel | first1 = Isaac |first2= Burton|last2=Randol| title=Eigenvalues in Riemannian geometry |series= Pure and Applied Mathematics|volume= 115| publisher = Academic Press| year= 1984 | isbn = 0-12-170640-0 |chapter= 
XI. The Selberg Trace Formula}} 
* {{cite journal| mr = 0612411 |last = Elstrodt | first = Jürgen | title = Die Selbergsche Spurformel für kompakte Riemannsche Flächen | lang = de | journal = Jahresber. Deutsch. Math.-Verein. |volume = 83 
| year = 1981 | pages = 45–77 | url = https://www.math.uni-bielefeld.de/JB_DMV/JB_DMV_083_2.pdf| location = Bielefeld |publisher = [[Deutsche Mathematiker-Vereinigung]]}}
* {{cite book | mr = 1483315 | last1=Elstrodt| first1= J. | last2 = Grunewald | first2= F. | last3= Mennicke| first3 = J. | title= Groups acting on hyperbolic space: Harmonic analysis and number theory| series = Springer Monographs in Mathematics | publisher = [[Springer-Verlag]]| year = 1998 | isbn = 3-540-62745-6}}
*{{Citation | last1=Fischer | first1=Jürgen | title=An approach to the Selberg trace formula via the Selberg zeta-function | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-15208-8 | doi=10.1007/BFb0077696 |mr=892317 | year=1987 | volume=1253}}
*{{Citation | author1-link=Israel Gelfand|last1=Gel'fand | first1=I. M. | last2=Graev | first2=M. I. | last3=Pyatetskii-Shapiro | first3=I. I. | title=Representation theory and automorphic functions | publisher=[[Academic Press]] | location=Boston, MA | series=Generalized Functions | isbn=978-0-12-279506-0 |mr=1071179 | year=1990 | volume=6}}
*{{cite book| mr = 0210827 | last =Godement| first = Roger | author-link = Roger Godement | chapter =
The decomposition of L<sup>2</sup>(G/Γ) for Γ=SL(2,Z)| year= 1966| title= Algebraic Groups and Discontinuous Subgroups | series = Proc. Sympos. Pure Math.| pages = 211–224 | publisher= [[American Mathematical Society]]| location = Providence}}
*{{Citation | author=[[Dennis Hejhal|Hejhal, Dennis A.]] | title=The Selberg trace formula and the Riemann zeta function | url=http://projecteuclid.org/euclid.dmj/1077311789 |mr=0414490 | year=1976 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=43 | issue=3 | pages=441–482 | doi=10.1215/S0012-7094-76-04338-6| url-access=subscription }}
*{{Citation | last1=Hejhal | first1=Dennis A. | title=The Selberg trace formula for PSL(2,R). Vol. I | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics, Vol. 548 | doi=10.1007/BFb0079608 |mr=0439755 | year=1976 | volume=548 | isbn=978-3-540-07988-0}}
*{{Citation | last1=Hejhal | first1=Dennis A. | title=The Selberg trace formula for  PSL(2,R). Vol. 2 | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-12323-1 | doi=10.1007/BFb0061302 |mr=711197 | year=1983 | volume=1001}}
* {{cite book | mr = 1942691 | last=Iwaniec | first= Henryk | author-link = Henryk Iwaniec | title= Spectral methods of automorphic forms | edition = Second | series = Graduate Studies in Mathematics | volume = 53 | publisher =[[American Mathematical Society]] | year=  2002 | isbn = 0-8218-3160-7 }}
* {{cite journal| mr =0555264 | last1= Lax| first1= Peter D.| author1-link= Peter Lax| last2= Phillips|first2 = Ralph S.| author2-link= Ralph S. Phillips| title = Scattering theory for automorphic functions| journal = [[Bull. Amer. Math. Soc.]]| volume= 2| year = 1980| pages = 261–295 | url = https://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14735-7/S0273-0979-1980-14735-7.pdf }}
*{{Citation | last1=McKean | first1=H. P. | author-link= Henry McKean| title=Selberg's trace formula as applied to a compact Riemann surface | doi=10.1002/cpa.3160250302  |mr=0473166 | year=1972 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=25 | pages=225–246 | issue=3}}
*{{Citation | last1=Selberg | first1=Atle | author-link=Atle Selberg|title=Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series |mr=0088511 | year=1956 | journal=J. Indian Math. Soc. |series=New Series | volume=20 | pages=47–87}}
*{{Citation | last1=Sunada | first1=Toshikazu | authorlink=Toshikazu Sunada |title=Trace formulae in spectral geometry| publisher=Springer-Verlag| series=Proc. ICM-90 Kyoto | year=1991 | pages=577–585}}

==External links==
*[http://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htm Selberg trace formula resource page]

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[[Category:Automorphic forms]]