By Jürgen Fischer

ISBN-10: 3540152083

ISBN-13: 9783540152088

The Notes provide an instantaneous method of the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) performing on the higher half-plane. the elemental notion is to compute the hint of the iterated resolvent kernel of the hyperbolic Laplacian so one can arrive on the logarithmic spinoff of the Selberg zeta-function. past wisdom of the Selberg hint formulation isn't assumed. the idea is built for arbitrary actual weights and for arbitrary multiplier structures allowing an method of recognized effects on classical automorphic varieties with out the Riemann-Roch theorem. The author's dialogue of the Selberg hint formulation stresses the analogy with the Riemann zeta-function. for instance, the canonical factorization theorem contains an analogue of the Euler consistent. eventually the final Selberg hint formulation is deduced simply from the homes of the Selberg zeta-function: this can be just like the method in analytic quantity idea the place the specific formulae are deduced from the houses of the Riemann zeta-function. except the fundamental spectral thought of the Laplacian for cofinite teams the ebook is self-contained and may be helpful as a short method of the Selberg zeta-function and the Selberg hint formulation.

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The Notes supply an immediate method of the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) performing on the higher half-plane. the elemental suggestion is to compute the hint of the iterated resolvent kernel of the hyperbolic Laplacian so that it will arrive on the logarithmic by-product of the Selberg zeta-function.

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**Additional info for An Approach to the Selberg Trace Formula via the Selberg Zeta-Function **

**Example text**

3 3 contains at least one element of For two distinct hyperbolic the associated ed above do not coincide if dj z J 6 K. = sup{l Then ] d ] J dj , log x} diameter of K. FA-]:3 ] WKj c B(zj, min(2,T) E j:1 N W'Kj) IKj,WKj I . A71U ! ) The last term is known to be equal to min(2"T) j=IE 2 ~(Kj) ( ) 2~ cosh(log x + 2dj)-1 : O(x) as This proves the lemma. X , ~ co . 3 Remark and Notation. is t h e unique number N(P) Assume that > I , such P that 6 F is h y p e r b o l i c , N(P) either I = A P A -] : DN(p) O A 6 SL(2,rR) .

It follows that for every hyperbolic m 6 ~ and j 6 {1,min(2,%)} the two sets Kj and P 6 F there exist , such that the hyperbolic U-mAjQA~IumKj Q 6 {P}F ' distance of satisfies: IKj , U-mA:] 1 UmK j I : IA~ 1 UmKj, Q A~ t UmKj I S log N(Q) = log N(P) as A71UmK. 3 3 contains at least one element of For two distinct hyperbolic the associated ed above do not coincide if dj z J 6 K. = sup{l Then ] d ] J dj , log x} diameter of K. FA-]:3 ] WKj c B(zj, min(2,T) E j:1 N W'Kj) IKj,WKj I . A71U ! ) The last term is known to be equal to min(2"T) j=IE 2 ~(Kj) ( ) 2~ cosh(log x + 2dj)-1 : O(x) as This proves the lemma.

A neigh- that {I,-I} h converges exist M 6 F ~ e-almost function. , ]~d×IH point bourhood and H(z,Mz') ' id v M6F~{I,-I ] As fixed x(M) ]M(Z') ) with f tr(Gkl(Z,z')Gk~(Z',Z))d~(z') • ~ (~(s+k)+~(s-k)-~,(a+k)-~(a-k)) I tr x(M) jM(z) H(z,Mz) M6F~{I,-I} every Theorem z 6 IH. 4) I X In- 1 n_>O + ½ I ) . ifn(Z)12 = -d • ~<~(s+k)+9(s-k)-9(a+k)-~(a-k) In - E M6 :'~{I tr x(M) jM(z) H(z,Mz)

### An Approach to the Selberg Trace Formula via the Selberg Zeta-Function by Jürgen Fischer

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