In mathematics, the Selberg trace formula, introduced by Selberg (1956), 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.
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In mathematics, the Selbergtraceformula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group...
non-commutative harmonic analysis, the idea is taken even further in the Selbergtraceformula, but takes on a much deeper character. A series of mathematicians...
theory into number theory, culminating in his development of the Selbergtraceformula, the most famous and influential of his results. In its simplest...
trace fromula Fischer, Jürgen (1987), An approach to the Selbergtraceformula via the Selberg zeta-function, Lecture Notes in Mathematics, vol. 1253,...
In mathematics, the local traceformula (Arthur 1991) is a local analogue of the Arthur–Selbergtraceformula that describes the character of the representation...
The Selbergtraceformula is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved that the Selberg zeta functions...
of the subject of group representations. See Selbergtraceformula and the Arthur-Selbergtraceformula for generalizations to discrete cofinite subgroups...
Urakawa, C. Gordon). In particular Vignéras (1980), based on the Selbergtraceformula for PSL(2,R) and PSL(2,C), constructed examples of isospectral,...
Hejhal works on analytic number theory, automorphic forms, the Selbergtraceformula and quantum chaos. From 1972 to 1974 he was a Sloan Fellow. In 1986...
theory of automorphic forms where it appears on one side of the Selbergtraceformula. Suppose f(x) is an absolutely continuous differentiable function...
conjecture Birch and Swinnerton-Dyer conjecture Automorphic form Selbergtraceformula Artin conjecture Sato–Tate conjecture Langlands program modularity...
Siegel modular forms. Important results in the theory include the Selbergtraceformula and the realization by Robert Langlands that the Riemann–Roch theorem...
years around 1960, in creating such a theory. The theory of the Selbergtraceformula, as applied by others, showed the considerable depth of the theory...
theorem Theorem of the three geodesics Curve-shortening flow SelbergtraceformulaSelberg zeta function Zoll surface Besse, A.: "Manifolds all of whose...
of the Selbergtraceformula, Journal of Soviet Mathematics, vol. 8, 1977, pp. 171–199 Spectral theory of automorphic functions, the Selberg zeta-function...
D. 1970), Wolf Prize medallist, mathematician known for Arthur-Selbergtraceformula and Arthur conjectures Jeffrey Brock (B.A. 1992), Dean of the School...
S L 2 ( Z ) . {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} ).} Selberg'straceformula shows that for an hyperbolic surface of finite volume it is equivalent...
automorphic forms: for example all modern treatments of the Arthur–Selbergtraceformula are done in this adélic setting. The modular group is usually defined...
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