Automorphic forms

Selberg trace formula

In mathematics, the Selberg trace formula, introduced by , 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. (Wikipedia).

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From playlist Using the Distance Formula / Midpoint Formula

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This video shows an example of determining the length of a segment on the coordinate plane by using the distance formula. Complete Video List: http://www.mathispower4u.yolasite.com or http://www.mathispower4u.wordpress.com

From playlist Using the Distance Formula / Midpoint Formula

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From playlist Algebra

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From playlist Partial Derivatives

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From playlist Algebra 2

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From playlist Joint IAS/PU Number Theory Seminar

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From playlist École d’été 2013 - Théorie des nombres et dynamique

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From playlist The Sato-Tate conjecture for abelian varieties

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From playlist Generalized Ramanujan Conjectures Applications by Peter Sarnak

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From playlist 2022 Summer School on the Langlands program

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Related pages

Modular group | Lie group | Martin Eichler | Ultrametric space | Group cohomology | Discrete group | Closed geodesic | Eichler–Shimura congruence relation | Poisson summation formula | Square-integrable function | Atle Selberg | Modular curve | Riemann surface | Congruence subgroup | Cusp form | Mathematics | Induced representation | Riemann zeta function | Selberg zeta function | Cocompact group action | Hecke operator | Spectrum (functional analysis) | Goro Shimura | Arithmetic geometry | Riemann–Roch theorem | Arthur–Selberg trace formula | Number theory | Prime number | Differential geometry | Laplace–Beltrami operator | Unitary representation | Eisenstein series | Resolvent formalism