Finding Riemann surfaces with several p-gonal morphisms

Abstract

Since every Riemann surface admits a meromorphic function then Riemann surfaces (algebraic curves) can be viewed as branched coverings of the Riemann sphere \(\mathbb {\hat{C}}\). A morphism from a Riemann surface on \(\mathbb {\hat{C}}\) of degree \(n\) is called an \(n\)-gonal morphism. If \(p\) is a prime and there is a \(p\)-gonal morphism from a Riemann surface \(X\) of genus \(g\) with \(g>(p-1)^{2}\) then the \(p\)-gonal morphism from \(X\) is unique. If the condition \(g>(p-1)^{2}\) is not fulfilled, examples of surfaces with more than one \(p\)-gonal morphism are known for the special case of cyclic coverings (see for instance [12]), here we shall construct examples for the case of irregular coverings. We deal with this less studied case in a two steps process: first we find several families of generic trigonal and dihedral pentagonal surfaces with more than one trigonal or pentagonal morphism, second we obtain families of Riemann surfaces of genera \((p-1)^2\), \((p-1)^{2}-2\), \((p-1)^{2}-3\) admitting several \(p\)-gonal morphims for every prime \(p>2\).