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 fundamental inspiration 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. past wisdom of the Selberg hint formulation isn't really assumed. the speculation is constructed for arbitrary genuine weights and for arbitrary multiplier structures allowing an method of recognized effects on classical automorphic types 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 consists of an analogue of the Euler consistent. ultimately the overall Selberg hint formulation is deduced simply from the houses of the Selberg zeta-function: this is often just like the strategy in analytic quantity concept the place the specific formulae are deduced from the homes of the Riemann zeta-function. except the elemental spectral thought of the Laplacian for cofinite teams the ebook is self-contained and may be important as a brief method of the Selberg zeta-function and the Selberg hint formulation.

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

**Example text**

The k n o w n [MOS], p. 2) lim (Gkl(Z,Z')-Gk~(Z,Z')):-~(~(s+k)+~(s-k)-~(a+k)-9(a-k) z~ z + lim ½ E z' ~ z usual By [El], normally 9 denotes section on O(z,Mz') and majorant of z', it is t h e r e f o r e for = a(1-a) s,a If z' 6 U z 6 IH test, ]M(Z') ~(z,Mz') -s is n o t F there > I , such all H(z,Mz') converges uniformly at every z 6 IH Re s, Re a > I, . Ikl-s, Thus, the by series Re s > I , (O(z,Mz')) ' in a n e i g h b o u r h o o d z' : z . a neigh- that {I,-I} h converges exist M 6 F ~ e-almost function.

2 v - I } many 2k elliptic IH As R : Rj - the p r i m i t i v e in to a r o t a t i o n {R} F the 62 0 cos sin | @ cos@ bitrary ing \ ! 8, tr X(R) also . Whenever denotes element depends R half R° on the is an ar- the o r d e r (depend- corresponding F-conjugacy to class only. Proof. As are not elliptic ~lell(Z) x(M) j M = x ( - M ) j _ M = fixed Z points Z (M £ F), we of have for all z 6 IH which F : tr X(S-IRS) JS_IRs (z) H(z, S-1RSz) ks(~(z, S-IRSz)) • {R} r S6Z (Rl\r 0<8<. The inner sum runs through right cosets ment R 6 {R} F .

R 2. 1 Lemma. Let k 6 IR, s 6 C, Re > ½ , O < e < ~ • Then 2 ei2kq°(cos %o)2~-2d