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 $p>2$.