By Jürgen Fischer

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 fundamental concept is to compute the hint of the iterated resolvent kernel of the hyperbolic Laplacian so one can arrive on the logarithmic by-product of the Selberg zeta-function. earlier wisdom of the Selberg hint formulation isn't assumed. the speculation is constructed for arbitrary actual weights and for arbitrary multiplier structures allowing an method of recognized effects on classical automorphic kinds with no 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 includes an analogue of the Euler consistent. ultimately the overall Selberg hint formulation is deduced simply from the houses of the Selberg zeta-function: this can be just like the technique in analytic quantity thought the place the categorical formulae are deduced from the homes of the Riemann zeta-function. except the elemental spectral concept of the Laplacian for cofinite teams the publication is self-contained and should be precious as a brief method of the Selberg zeta-function and the Selberg hint formulation.

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Extra resources for An Approach to the Selberg Trace Formula via the Selberg Zeta-Function

Sample text

1. [Ro 2], This the in not of the proves Ejp(z, lie s o >_ ½ of s ~ ½} series all of t h e m Re Re ) ]½,1]. all the series assertion. D 2. 7), Theorem r . 10) and Expansion Theorem a base for the proof of the trace formula. 4 are with we consider the integral (l-~) S tr(Gkl (z,z') Gkp (z' ,z)) d~(z') , z 6 IH F (tr A :: trace of the square matrix A ). 4 into m. 1. E3 ~1 7 ~ T~ ~ j I p=] -~\¼+t--i i . ~I ~)" (z'21+'It)12dt ~ n 2IEjp On the other hand, we have from the resolvent equa- tion: (I-P) S trO ~n -~ t r < G k l ( Z , Z ' ) - ~ p (z,z')) .

5 we see that ~n-~ Ifn(Z)i tr(%(zz) Z~-~ Z m, - Z j=Zl p=1 is ~-integrable resolvent kernel. 4. 1 Theorem. I = s(1-s), OF THE TRACE BY THE EIGENVALUES Let U = a(1-a) . 4 with corresponding eige~nvalues = Z S tr(Gkl(W,Z') F { I -~k s,a 6 C , Re s, Re a > I, Ik l - s , mal system of eigenfunctions (l-U) OF that (fn)n>_O -Ak according is a maximal r orthonor- to Expansion (In)n>O • Then for all Gku(Z',Z)) Ikl-a { INO z,w 6 IH Theorem we have: do(z') I_ )' In U n_>O \ In-I m. j I p=1 Proof.

296, p. ) F,G: C ,¢ , X where e is h o l o m o r p h i c and (s 6 ¢) constant C > O . , p. 130, entire for all ~ e x p ( C l s l 4) sufficiently (Cf. coincide. holds. IG(s) I ~ e x p ( C l s l 4) some s that ~(s) for and on in the of z C IH is the of V the c h o i c e theory of ; (cf. square of by v i r t u e of the [E2], the o p e r a t o r of R e m a r k fundamental integral Kor. b)). 9 domain (cf. of the the Z of [RSzN], endomor- integral does F . ) now a 38 follows the existence eigenfunctions real a n d of of a c o u n t a b l e of the o p e r a t o r finite Every function -~k: multiplicity, non-zero-eigenvalues orthonormal Dk the system ~ H k .