An Approach to the Selberg Trace Formula via the Selberg by Jürgen Fischer

By Jürgen Fischer

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 proposal is to compute the hint of the iterated resolvent kernel of the hyperbolic Laplacian on the way to arrive on the logarithmic by-product of the Selberg zeta-function. prior wisdom of the Selberg hint formulation isn't assumed. the speculation is built for arbitrary genuine weights and for arbitrary multiplier structures allowing an method of identified 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 consists of an analogue of the Euler consistent. eventually the overall Selberg hint formulation is deduced simply from the houses of the Selberg zeta-function: this is often just like the technique in analytic quantity conception the place the specific formulae are deduced from the homes of the Riemann zeta-function. except the fundamental spectral conception of the Laplacian for cofinite teams the ebook is self-contained and may be invaluable as a short method of the Selberg zeta-function and the Selberg hint formulation.

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J E {1,min(2,T) Select } , such S E F , such that = ax(Q) by m = O one , c KI end j : I. point, Consequently, ~ Yl +I not real exceeding and there that is o n e exists . Sz' Then Re A I S z ' . 6 A~IUmKI proved (by c h o i c e of < g . Im A I Sz Im A 2 S z unique norm . As z' 6 a x ( P ) , N AIIumK I . the Im A I S Z least < Im AISZ' x ]y1,Y1+1] 6 ax(Q and . of largest and is m e t at the sub- S Yl +I has be is a c o m p a c t z 6 IH} the 6 S ax(P) ax(P) m I} trace points £ ~ a x ( A I Q A ~ I) = A I S Let on z 6 ax(P) c K I .

5: {s 6 ¢: Re s > I} uniformly in , { , -s-m on c o m p a c t sets in Re s > I} , and S Gklhyp(Z) F I d~0(z) : 2 s - ~ -hyp (s) -hyp ~ N (Po) -ns X X tr X(Po )n log N(P o) • 2s-I {po]F n:1 I-N(Po )-n I Re s > I, Ikl-s { INo, I = s(1-s). 57 By a n a l o g y tion, by ~ -hyp the Euler is c a l l e d product expansion the S e l b e r q of the zeta-function Riemann and zeta-func- frequently denoted Z . 2,3 THE C O N T R I B U T I O N OF THE ELLIPTIC ELEMENTS During F with the to the hand computation trace in the there following of the contribution will occur two lemmata.

T E(z,s;Vjp,Aj,k,x) will always be . _P_roposition. ) we adopt The zeroth Fourier ~V the following result. coefficient of the expansion of with respect Ejp(Z,s) to has the form I pjp,l(s)y 1-s + 6jl VlpY s , if mI > O if mI = O . Ujp,l(Y,S O Here Pjp,1 is a meromorphic least at the points {s E ~: Re s : ½} Moreover, nate , function of holomorphy (6jl: Kronecker the following Dirichlet of on ~ Ejp(Z, which is holomorphic ), especially at on the line symbol). ,Y ; p:1 ..... mj ; q=1 ..... 7 q 6 Proposition.

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