Galois Closure of a Fivefold Covering and Decomposition of Its Jacobian

Abstract

For an arbitrary fivefold ramified covering \(\varvec{f :X\rightarrow Y}\) between compact Riemann surfaces, each possible Galois closure \(\varvec{\hat{f}:\hat{X}\rightarrow Y}\) is determined in terms of the branching data of \(\varvec{f}\). Since \(\varvec{{{\,\textrm{Mon}\,}}(f)}\) acts on \(\varvec{\hat{f}}\), it also acts on the Jacobian variety \(\varvec{{{\,\textrm{J}\,}}(X)}\), and we describe its group algebra decomposition in terms of the Jacobian and Prym varieties of the intermediate coverings of \(\varvec{\hat{f}}\). The dimension and induced polarization of each abelian variety in the decomposition is computed in terms of the branching data of \(\varvec{f}\).