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In the first half of the present paper, we study higher-level generalizationsof differential modules in positive characteristic. These objects may beregarded as ring-theoretic counterparts of vector bundles on a curve equippedwith an action of the ring of (logarithmic) differential operators of finitelevel introduced by P. Berthelot (and C. Montagnon). The existence assertionfor a cyclic vector of a differential module is generalized to higher levelunder mild conditions. In the second half, we introduce (dormant) opers oflevel $N > 0$ on a pointed smooth curve whose structure group is either$\mathrm{GL}_n$ or $\mathrm{PGL}_n$. Some of the results on higher-leveldifferential 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 oflevel $N$. Finally, in the case where the underlying curve is a $3$-pointedprojective line, we establish a bijective correspondence between dormant$\mathrm{PGL}_2$-opers of level $N$ and certain tamely ramified coverings.

Differential modules and dormant opers of higher level