Abstract
This paper may be considered as a preparation for later investigations. I shall prove some theorems and lemmas which are of general interest to the theory of modular functions of several variables. Some of the results were already known but it seemed desirable to find better proofs. The part about Poincare's series is closely connected with some recent work of H. Klingen [19], [20], [21]; a more detailed explanation about this connection may be found at the end of Chapter III, ? 3. The paper consists of three chapters, the contents of which is the following. The first chapter deals with the symplectic group. Some inequalities and the boundedness of certain expressions will be proved. The second chapter is devoted to Hilbert-Siegel's modular groups. First we define arithmetically certain equivalence classes and prove that there are only finite many such classes provided some definite expression is bounded. These results are needed to prove the convergence of Poincare's series. Then we deal with elliptic fixed points of Hilbert-Siegel's modular groups and improve some result obtained in [11]. At the end of the second chapter I prove that the quotient spaces of the generalized upper half-plane with respect to the full Siegel's modular group and the so called group of integral modular substitutions have the first homology group zero provided n > 1. In chapter three we show that under certain assumptions a HilbertSiegel's modular form of weight zero is constant. Then we investigate the convergence of certain integrals and in connection herewith the uniform convergence of Poincare's series. Finally some theorem about separation of variables through modular functions is deduced. This paper was partially supported by an N. S. F. grant.
Published Version
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