Differential modules and dormant opers of higher level

The aim of the present paper is to develop the theory of $\mathcal{D}$-modules in positive characteristic. In Sections 2, 3, and 4, we study higher-level generalizations of differential modules in positive characteristic. These objects may be regarded as ring-theoretic counterparts of vector bundles on an algebraic curve equipped with an action of the ring of (logarithmic) differential operators of finite level introduced by P. Berthelot and C. Montagnon. The well-known existence assertion for a cyclic vector of a differential module is generalized to higher level. In Sections 5, 6, and 7, we introduce and discuss (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 in Sections 3 and 4 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. These assertions are building blocks to establish the enumerative geometry of higher-level dormant opers.