Jrgen Fischer;Ja1;4rgen Fischer
Release Date: 23 April 1987
Format: Paperback / softback
Pages: 188
Category:
Number TheoryPublisher:
SpringerISBN: 9783540152088
ISBN10: 3540152083
The Notes give a direct approach to the Selberg zetafunction for cofinite discrete subgroups of SL (2,#3) acting on the upper halfplane. The basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic Laplacian in order to arrive at the logarithmic derivative of the Selberg zetafunction. Previous knowledge of the Selberg trace formula is not assumed. The theory is developed for arbitrary real weights and for arbitrary multiplier systems permitting an approach to known results on classical automorphic forms without the RiemannRoch theorem. The author's discussion of the Selberg trace formula stresses the analogy with the Riemann zetafunction. For example, the canonical factorization theorem involves an analogue of the Euler constant. Finally the general Selberg trace formula is deduced easily from the properties of the Selberg zetafunction: this is similar to the procedure in analytic number theory where the explicit formulae are deduced from the properties of the Riemann zetafunction. Apart from the basic spectral theory of the Laplacian for cofinite groups the book is selfcontained and will be useful as a quick approach to the Selberg zetafunction and the Selberg trace formula.
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