On \(p\)-hyperelliptic involutions of Riemann surfaces (Q2568871)

A compact Riemann surface \(X\) of genus \(g\geq 2\) is said to be \(p\)-hyperelliptic if it admits a conformal involution \(\rho \) such that the quotient \(X/\rho \) has genus \(p\). The involution \(\rho\) is called a \(p\)-hyperelliptic involution or \(p\)-involution, in short. A \(p\)-involution has \(2g+2-4p\) \ fixed points. In \textit{E. Tyszkowska}, ''On \(pq\) -hyperelliptic Riemann surfaces''. Colloq. Math. 103, No. 1, 115-120 (2005; Zbl 1080.30037)] surfaces which admit \(p\)-hyperelliptic and \(q\)-hyperelliptic involutions were studied. In such a case \(X\) is called a \(pq\)-hyperelliptic surface. In the paper under review, all involutions are \(p\)-involutions. Let us suppose that \(X\) is \(p\)-hyperelliptic. It is known that if \(g>4p+1\), then \(\rho\) is unique and central in the group of all automorphisms of \(X\). The author studies how many \(p\)-hyperelliptic involutions are admitted by \(X\) according to the range of the genus \(g\). She proves that for \(3p+2\leq g \leq 4p+1\) every two \(p\)-involutions commute and if \(g\neq 3p+2\), \(X\) can admit two and no more such involutions. The case \(g=3p+2\) gives different possibilities. \(X\) can admit at most the following numbers of \(p\)-involutions: \(2\) (if \(p\equiv 0(2)\) or \(p\equiv 3(4)\)), \(3\) (if \(p\equiv 1(4)\) and \(p>5\)), \(5\) (if \(p=1\)) and \(6\) (if \(p=5\)). For \(2p-1<g< 3p+2\), several bounds are given on the number of pairwise commuting \(p\)-involutions. Finally, if \(G\) is the automorphism group of order \(2N\) of a \(p\)-hyperelliptic Riemann surface \(X\) of genus \(g\geq 2\) and \(g \neq 2p-1\), then an upper bound is given for the number of central \(p\)-involutions of \(X\). This bound depends on \(N,p,g\) and the multiplicities of the points of ramification of the canonical projection \(X\rightarrow X/G\).