Differential forms on the curves associated to Appell-Lauricella hypergeometric series and the Cartier operator on them

The curve C over C associated to Appell-Lauricella hypergeometric series and regular differential forms on its desingularization were previously studied by Archinard. In this paper, we first generalize Archinard’s results for a field K under a mild condition on its characteristic. Second, we describe a partial desingularization of C and the space of global sections of its dualizing sheaf, especially we give an explicit basis of it. Finally, when the characteristic is positive, we show that the Cartier operator on the space can be defined and describe it in terms of Appell-Lauricella hypergeometric series.