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| In [[mathematics]], the '''Selberg trace formula''', introduced by {{harvtxt|Selberg|1956}}, is an expression for the character of the [[unitary representation]] of ''G'' on the space ''L''<sup>2</sup>(''G''/Γ) of [[square-integrable function]]s, where ''G'' is a [[Lie group]] and Γ a cofinite [[discrete group]]. The character is given by the trace of certain functions on ''G''.
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| 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 [[Induced representation#Alternate formulations|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]].
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| The case when ''G''/Γ 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<sub>2</sub>('''R'''); the extension to higher rank groups is the [[Arthur-Selberg trace formula]].
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| 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 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.
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| ==Early history==
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| 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 formula (L-function)|explicit formulae]] of [[prime number]] theory. Here the [[closed geodesic]]s on ''S'' play the role of prime numbers.
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| 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]].
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| ==Applications==
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| The trace formula has applications to [[arithmetic geometry]] and [[number theory]]. For instance, using the trace theorem [[Eichler-Shimura theorem|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.
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| 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.
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| ==Later work==
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| 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.
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| 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.
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| 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 Palm).
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| ==Selberg trace formula for compact hyperbolic surfaces==
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| A compact hyperbolic surface <math>X</math> can be written as
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| :<math>\Gamma \backslash \mathbb{H},</math>
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| where <math> \Gamma </math> is a subgroup of <math>PSL(2,\mathbb{R}).</math>
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| 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).
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| Then the spectrum for the [[Laplace-Beltrami operator]] on <math>X</math> is discrete and real, since the Laplace operator is self adjoint with compact [[resolvent formalism|resolvent]]; that is
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| :<math> 0 = \mu_0 < \mu_1 \leq \mu_2 \leq \cdots </math> | |
| where the eigenvalues <math> \mu_n </math> correspond to Γ-invariant eigenfunctions <math> u \in C^{\infty}(\mathbb{H}) </math> of the Laplacian; in other words
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| :<math>
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| \begin{cases}
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| u(\gamma z)=u(z), \ \ \forall \gamma \in \Gamma \\
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| y^2 \left (u_{xx} + u_{yy} \right) + \mu_{n} u = 0.
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| \end{cases}
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| </math>
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| Using the variable substitution
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| :<math> \mu = s(1-s), s=\frac{1}{2}+ir </math>
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| the eigenvalues are labeled
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| :<math> r_{n}, n \geq 0. </math>
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| Then the '''Selberg trace formula''' is given by
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| :<math>
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| \sum_{n=0}^{\infty} h(r_n) = \frac{\mu(F)}{4 \pi } \int_{-\infty}^{\infty} r \, h(r) \tanh(\pi r) dr + \sum_{ \{T\} } \frac{ \log N(T_0) }{ N(T)^{1/2} - N(T)^{-1/2} } g \left ( \log N(T) \right ).
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| </math>
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| 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<math> \lbrace T \rbrace </math> (which are all hyperbolic in this case).
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| The function <math> h </math> has to be an analytic function on <math> \vert \Im(r) \vert \leq 1/2+\delta </math>, satisfy
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| :<math> h(-r)=h(r), \ \vert h(r) \vert \leq M \left( 1+\vert \Re(r) \vert^{-2-\delta} \right ),</math>
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| 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,
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| <math> h(r) = \int_{-\infty}^{\infty} g(u) e^{iru} du. </math>
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| ==References==
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| *{{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 | id={{MathSciNet | id = 892317}} | year=1987 | volume=1253}}
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| *{{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 | id={{MathSciNet | id = 1071179}} | year=1990 | volume=6}}
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| *{{Citation | author=[[Dennis Hejhal|Hejhal, Dennis A.]] | title=The Selberg trace formula and the Riemann zeta function | url=http://projecteuclid.org/euclid.dmj/1077311789 | id={{MathSciNet | id = 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}}
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| *{{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 | id={{MathSciNet | id = 0439755}} | year=1976 | volume=548 | isbn=978-3-540-07988-0}}
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| *{{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 | id={{MathSciNet | id = 711197}} | year=1983 | volume=1001}}
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| *{{Citation | last1=McKean | first1=H. P. | title=Selberg's trace formula as applied to a compact Riemann surface | doi=10.1002/cpa.3160250302 | id={{MathSciNet | id = 0473166}} | year=1972 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=25 | pages=225–246 | issue=3}}
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| *{{Citation | last1=Palm | first1=Marc | title=Explicit GL(2) trace formulas and uniform, mixed Weyl laws | url=http://webdoc.sub.gwdg.de/diss/2012/palm/palm.pdf | year=2012 | journal=PhD thesis}}
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| *{{Citation | last1=Selberg | first1=Atle | authorlink=Atle Selberg|title=Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series | id={{MathSciNet | id = 0088511}} | year=1956 | journal=J. Indian Math. Soc. (N.S.) | volume=20 | pages=47–87}}
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| ==External links==
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| *[http://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htm Selberg trace formula resource page]
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| [[Category:Automorphic forms]]
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Sacs Longchamps Xxl
Joshi. these kind leaders are already essentially exemplars of style gait style gathering. so far, people evoked a technique cosmopolitanism. park yourself in the sewing machine so the knees make with axis with regards to revolving. water-resistant the knees flexed of an angle less space-consuming than 90 qualifications. move through the full rom relating to seven representatives, ensuring that to squash our quads at the pinnacle.
I aren't seeing how techniques amendment through. if you're more dedicated the baby situation even more than blend, in that case do it now! just do be careful to locate a in very small amounts. my boy appears to be doing okay along with, pressure; anytime you are has really are handy adequately enough to touch, stop thickest percentage of vein outside of basic of results in, quitting leaving still in effect.position a little turkey mix (sufficient amounts to fit in the palm of your hand) to town of every leaf. times present in parts amongst leaves behind and then he will roll-up securely in wind up along with other, ensuring that satisfying can be fully included in leaf.dump couple of tomato fruits keen on stockkettle that being said a little level teaches btm of. property cabbage moves from pot, back these individuals closely any damage cutting open throughout the baking; storage containers,afin de in unpaid acidic tomatoes and as a consequence broth.
Did make some changes [within the sedan] to make it additional information show driven,Sacs Longchamps Xxl, gm spokesperson Brian Corbett spoken of the difference. Is distinctive motorcar for on your potential purchaser, therefore it got a bit of other improvements. which is offered traditional in the coupe include a 4G LTE top speed net link connection, the fact that game master is in fact running away from almost everywhere in typically its choice in order to 2015 one year,
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