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An Elementary Proof of a Fixed Point Theorem of J. Lewittes and D. L. McQuillan

Published online by Cambridge University Press:  20 November 2018

Arthur K. Wayman
Arthur K. Wayman*
Affiliation:
Department of Mathematics, California State University, Long Beach Long BeachCA 90840
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In (3), J. Lewittes establishes a connection between the number of fixed points of an automorphism of a compact Riemann surface and Weierstrass points on the surface; Lewittes′ techniques are analytic in nature. In (4), D. L. McQuillan proved the result by purely algebraic methods and extended it to arbitrary algebraic function fields in one variable over algebraically closed ground fields, but with restriction to tamely ramified places. In this paper we will give a different proof of the theorem and show that it is an elementary consequence of the Riemann-Hurwitz relative genus formula. Moreover, we can remove the tame ramification restriction.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 1 , 01 March 1978 , pp. 99 - 101
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Boseck, H., “Zur Théorie der Weierstrasspunkte,” Math. Nachr. 19 (1958), 29-63.Google Scholar
2. Chevalley, C., Introduction to the Theory of Algebraic Functions of One Variable, Amer. Math. Soc, New York, 1951.Google Scholar
3. Lewittes, J., “Automorphisms of compact Riemann surfaces,” Amer. J. Math., 85 (1963), 734-752.Google Scholar
4. McQuillan, D. L., “A note on Weierstrass points,” Canad. J. Math., 19 (1967), 268-272.Google Scholar
5. Schmidt, F. K., “Zur arithmetischen Theorie der algebraischen Funktionen. II. Allgemeine Théorie der Weierstrasspunkte” Math. Zeit. 45 (1939), 75-96.Google Scholar