Cyclic \'{e}tale coverings of generic curves and ordinariness of dormant opers

Abstract

The ordinariness of elliptic curves is essential in proving various expected properties of elliptic curves in positive characteristic and can be extended to algebraic curves of arbitrary genus. The present paper deals with another kind of extension, i.e., ordinariness of dormant $\mathfrak{sl}_2$-opers, or more generally, dormant $\mathfrak{sl}_n$-opers. We prove that, for dormant $\mathfrak{sl}_2$-opers on elliptic curves, this notion is essentially equivalent to the classical ordinariness. Moreover, the main result of the present paper asserts that the pull-back of an ordinary dormant $\mathfrak{sl}_n$-oper on a general curve by a cyclic covering is ordinary whenever the order of its Galois group is prime to the characteristic of the base field. This result may be regarded as an analogue of a result by S. Nakajima for ordinary algebraic curves.