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ON WIMAN BOUND FOR ARITHMETIC RIEMANN SURFACES

Published online by Cambridge University Press:  01 May 2003

MIKHAIL BELOLIPETSKY  and
GRZEGORZ GROMADZKI
MIKHAIL BELOLIPETSKY
Affiliation:
Sobolev Institute of Mathematics, Koptyuga 4, 630090 Novosibirsk, Russia, Max Planck Institute of Mathematics, Vivatsgasse 7, 53111 Bonn, Germany e-mail: [email protected]
GRZEGORZ GROMADZKI
Affiliation:
Institute of Mathematics University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland e-mail: [email protected]
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Abstract

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We show that the order of an automorphism of an arithmetic Riemann surface of genus $g$ is not greater than $2g\,{-}\,2$, provided $g$ is large enough. This bound is an arithmetic analog of the classical Wiman bound. We prove that it is sharp and attained for any genus but in contrast to the general case the automorphisms of maximal order act without fixed points. This allows us to consider the automorphisms which act on arithmetic Riemann surfaces and have a given number of fixed points. For these automorphisms we describe the asymptotic behavior of their orders.

Keywords

30F1014H45
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 45 , Issue 1 , January 2003 , pp. 173 - 177
Copyright
2003 Glasgow Mathematical Journal Trust