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By Andreas Juhl

Dynamical zeta services are linked to dynamical platforms with a countable set of periodic orbits. The dynamical zeta features of the geodesic circulation of lo­ cally symmetric areas of rank one are identified additionally because the generalized Selberg zeta capabilities. the current publication is anxious with those zeta services from a cohomological perspective. initially, the Selberg zeta functionality seemed within the spectral idea of automorphic varieties and have been instructed through an analogy among Weil's particular formulation for the Riemann zeta functionality and Selberg's hint formulation ([261]). the aim of the cohomological conception is to appreciate the analytical houses of the zeta capabilities at the foundation of compatible analogs of the Lefschetz mounted aspect formulation during which periodic orbits of the geodesic stream take where of mounted issues. This method is parallel to Weil's notion to investigate the zeta features of professional­ jective algebraic types over finite fields at the foundation of appropriate models of the Lefschetz fastened element formulation. The Lefschetz formulation formalism exhibits that the divisors of the rational Hassc-Wcil zeta capabilities are decided by way of the spectra of Frobenius operators on l-adic cohomology.

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A of. 45). We shall see that for currents e which are not constant along P- but P- -strongly harmonic (with at most polynomial growth) there are corresponding solution formulas involving higher order polynomials of D. In Chapter 4 we introduce all basic operators and complexes on SY and SX and study their elementary properties. In Chapter 5 and Chapter 6 we give a detailed analysis of those complexes which are relevant for the theory of the Selberg zeta function Zs. For the analysis of the differential operators we shall use their models in Iwasawa coordinates (the compact or Iwasawa model) and Bruhat coordinates (the w= 30 Chapter 1.

89) q of D(Zs), where mq is the dimension of the generalized eigenspace of the transfer operator 'cq('\) (depending on a Markov partition) for the eigenvalue 1. 5). 89) massive cancellations take place. + and D). , ·)sx on the spaces of currents being responsible for the zeros and poles of Z s. 7/(i),(ii) no cancellation takes place. , for some ,\ E C non-trivial contributions sum up to O. 3. :£.... :£.... :£.... 91 ) of CDO-forms to the tangential complex 0---+ n(n-l,O) (SX) d- --+ n(n-2,O)(SX) d- d- --+ ...

W =0 = 0. It follows that 'A 'A 'A 'A CH(1,o)(SX) ~ C:H(1,O) (SX), CH(O,O) (SX) ~ C:H(O,O) (SX) and thus the identities . 'A . ~ . 'A ordA(Zs) = dIm CH(1,O) (SX) - dImCH(o,o) (SX) ordA(Zs) A = dImCJ-C(l,O) (SX) . 'A - dIm CJ-C(O,O) (SX) are equivalent restatements of the above formula for ord A (Z s). The decomposition 'A _ 'A CC(O,O) (SX) = d CC(1,O) (SX), Ai- 0 is a consequence of the identity , _ + OW = d D. 88) for a constant c i- 0. 46)) implies ' + w) = D. +'Ow = cAD. +w, wE CC(O,O)(SX), 'A O(D.

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