Differential modules and dormant opers of higher level

In the first half of the present paper, we study higher-level generalizations of differential modules in positive characteristic. These objects may be regarded as ring-theoretic counterparts of vector bundles on a curve equipped with an action of the ring of (logarithmic) differential operators of finite level introduced by P. Berthelot (and C. Montagnon). The existence assertion for a cyclic vector of a differential module is generalized to higher level under mild conditions. In the second half, we introduce (dormant) opers of level $N > 0$ on a pointed smooth curve whose structure group is either $\mathrm{GL}_n$ or $\mathrm{PGL}_n$. Some of the results on higher-level differential modules are applied to prove a duality theorem between dormant $\mathrm{PGL}_n$-opers of level $N$ and dormant $\mathrm{PGL}_{p^N-n}$-opers of level $N$. Finally, in the case where the underlying curve is a $3$-pointed projective line, we establish a bijective correspondence between dormant $\mathrm{PGL}_2$-opers of level $N$ and certain tamely ramified coverings.