Selberg trace formula for compact quotient. Selberg also established trace formulas for noncompact quotients of rank 1, and our formula can be regarded as an 

22 Sep 2005 Eisenstein Series and the Selberg Trace Formula. II. STOR. H. Jacquet; D. Zagier . Transactions of the American Mathematical Society, Vol. 300 

The character is given by the trace of certain functions on G. In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL 2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of G (A) on the discrete part L2 Selberg Trace Formula Let run over all distinct primitive ordered periodic geodesics, and let denote the positive length of, then every even function analytic in and such that for satisfies the summation formula where is the genus of the surface whose area is by the Gauss-Bonnet formula. tum systems, the role of Selberg’s formula is taken over by the semiclassical Gutzwiller trace formula [11], [7]. We begin by reviewing the trace formulas for the simplest compact man-ifolds, the circle S1 (Section 1) and the sphere S2 (Section 2). In both cases, the corresponding geodesic flow is integrable, and the trace formula is a con- The Selberg trace formua is an expression for certain sums of eigenvalues of the Laplace operator on a compact hyperbolic Riemann surface (recalled e.g.

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42. 15. Weyl's law and the existence of cusp forms. 45. on Γ\H2.

2018-10-20 · In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle of the computation of a residue. We solve this puzzle and complete the derivation. The main input is an extension of the theory of regularized integrals invented by Zagier, which is of

We begin by reviewing the trace formulas for the simplest compact man-ifolds, the circle S1 (Section 1) and the sphere S2 (Section 2). In both cases, the corresponding geodesic flow is integrable, and the trace formula is a con- The Selberg trace formua (Selberg 56) is an expression for certain sums of eigenvalues of the Laplace operator on a compact hyperbolic Riemann surface (recalled e.g. as Bump, theorem 19).

The Selberg Trace Formula in Mathematical Physics. Applications and Generalizations . Quantum Chaos. The Selberg Zeta-Function and the Riemann ζ-Function. String Theory. The Selberg Trace Formula on Symmetric Space Forms of Rank One. The Selberg Trace Formula on D-Dimensional Hyperbolic Space. Cosmology. Trace Formulæ on Spheres

In §3 we apply the sieving process to pass from the full space to newforms, and then to twist-minimal forms.

In both cases, the corresponding geodesic flow is integrable, and the trace formula is a con- The Selberg trace formua (Selberg 56) is an expression for certain sums of eigenvalues of the Laplace operator on a compact hyperbolic Riemann surface (recalled e.g. as Bump, theorem 19). holdsforanyF-rationalcharacterofG.Set H G:G=G1 ×A G −→proj A G −→log a G. ForaparabolicsubgroupP=MU,wewriteF(M),P(M)andL(M)forthesetof parabolic subgroups containingM, the setofparabolic subgroups havingM as aLevi THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES IGOR PROKHORENKOV 1. Introduction to the Selberg Trace Formula This is a talk about the paper H. P. McKean: Selberg’s Trace For-mula as applied to a compact Riemann surface (1972).
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It describes the character of the representation of G (A) on the discrete part L2 Selberg Trace Formula Let run over all distinct primitive ordered periodic geodesics, and let denote the positive length of, then every even function analytic in and such that for satisfies the summation formula where is the genus of the surface whose area is by the Gauss-Bonnet formula. Selberg’s Trace Formula: An Introduction 3 ρ −ρ Reρ Im ρ ’ ’ we find X m∈Z (ρ2 −m2)−1 = πi ρ X n∈Z e−2πi|n|ρ. (8) The right hand side resembles the geometric series expansion of cotz for Imz<0, cotz= 2ie−izcosz 1−e−2iz = i(1+e−2iz) X∞ n=0 e−2inz= i X n∈Z e−2i|n|z. (9) Hence X m∈Z (ρ2 −m2)−1 = π ρ cot(πρ), (10) 2.
Selberg trace formula

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The character is given by the trace … Shimura varieties and the Selberg trace formula 2 If we follow this suggestion, we might divide the problem into three parts. (a) If a reductive group H over Q is given, together with an automorphic form πon it, or rather what has recently been called an automorphic representation, as well as a representation r of the associate group LH, then one may introduce an L-function L(s,π,r), which THE SELBERG TRACE FORMULA. V 353 (8) 3p > 0 st Va £ CP(G) st XF *a = a or a*xF = a, (7(a) is trace class. (The proof may be found in TES, pp. 375-376.) Let 0 be the universal enveloping algebra of the complexification of the Lie algebra g of G. Fix a positive semidefinite, elliptic element A in