Automorphisms Of Compact Riemann Surfaces Research Articles

Fixed Points of Automorphisms of Compact Riemann Surfaces

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.

Automorphisms of compact Riemann surfaces and the vanishing of theta constants

Automorphisms of compact Riemann surfaces and the vanishing of theta constants

Abelian Groups of Automorphisms of Compact Riemann Surfaces

Proceedings of the London Mathematical SocietyVolume s3-15, Issue 1 p. 699-712 Articles Abelian Groups of Automorphisms of Compact Riemann Surfaces C. Maclachlan, C. Maclachlan The University, BirminghamSearch for more papers by this author C. Maclachlan, C. Maclachlan The University, BirminghamSearch for more papers by this author First published: 1965 https://doi.org/10.1112/plms/s3-15.1.699Citations: 45AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes3-15, Issue11965Pages 699-712 RelatedInformation

Automorphisms of Compact Riemann Surfaces

Automorphisms of Compact Riemann Surfaces