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Siegel's mean value theorem in the geometry of numbers

Published online by Cambridge University Press:  24 October 2008

A. M. Macbeath  and
C. A. Rogers
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Extract

Our main object is to give an exposition of a simplified version of Siegel's proof of his mean value formula in the Geometry of Numbers, also proved by A. Weil in a more general context ((1), (2), (3), (4)). In deriving the particular case which is Siegel's formula, however, certain difficulties of convergence arise, which were resolved by Siegel through rather tedious calculations, and by Weil through the use of Fourier analysis in topological groups. A new method of dealing with these difficulties, depending on Lebesgue's bounded convergence theorem and the relatively simple Lemma 8, seems to be our major contribution. In other respects our proof is very similar to Siegel's. At some points we have simplified his treatment; at others, where his work seems rather condensed, we have expanded it. We are indebted to Mahler (9) for some of the simplifications and grateful to him for letting one of us see his unpublished work. By using Lemma 8 and giving a separate proof of the existence of a fundamental domain for the unimodular group, we give a proof which is independent of Minkowski's reduction theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 2 , April 1958 , pp. 139 - 151
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCES

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