On periodic maps over surfaces with large periods (Q973686)

A long-standing theorem due to Wiman (1895) states that the order of a conformal automorphism of a compact Riemann surface (or complex algebraic curve) of genus \(g > 1\) is at most \(4g+2\). This bound is sharp, and in fact is attained for all such \(g\). Wiman's theorem was extended by \textit{R. S. Kulkarni} [From the proceedings of the AMS special session with related papers, January 4--5, 1995, San Francisco, CA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 201, 63--79 (1997; Zbl 0863.30050)] to classify such surfaces admitting an automorphism of order \(4g\) or more. In the current paper, the author extends this further to determine all cases where the surface admits an automorphism of order \(3g\) or more, for \(g > 2\). This can be achieved using the Riemann-Hurwitz formula and application of \textit{W. J. Harvey}'s theory of cyclic groups of automorphisms of compact Riemann surfaces [Q. J. Math., Oxf. II. Ser. 17, 86--97 (1966; Zbl 0156.08901)]. There are four infinite families, as follows: (a) order \(4g+2\), signature \((0; 2,2g+1,4g+2)\), for all \(g\); (b) order \(4g\), signature \((0; 2,4g,4g)\), for all \(g\); (c) order \(3g+3\), signature \((0; 3,g+1,3g+3)\), for all \(g \not\equiv 2\) mod \(3\); (d) order \(3g\), signature \((0; 3,3g,3g)\), for all \(g\); plus five sporadic cases (viz.~orders 12, 20, 28, 30 and 36, for \(g = 4,6,9,10\) and \(12\) respectively). The result can easily be extended to cover also the case of genus \(2\), by the addition of one more sporadic case (viz.~order \(6\), with signature \((0;2,2,3,3)\)).