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In the present paper, we study the p-adic Teichmuller theory in the case where p = 3. In particular, we discuss nilpotent admissible/ordinary indigenous bundles over a projective smooth curve in characteristic three. The main result of the present paper is a characterization of the supersingular divisors of nilpotent admissible/ordinary indigenous bundles in characteristic three by means of various Cartier operators. By means of this char- acterization, we prove that, for every nilpotent ordinary indigenous bundle over a projective smooth curve in characteristic three, there exists a connected nite etale covering of the curve on which the indigenous bundle is not ordinary. We also prove that every projective smooth curve of genus two in characteristic three is hyperbolically ordinary. These two applications yield negative, partial positive answers to basic questions in the p-adic Teichmuller theory, respectively.

Nilpotent admissible indigenous bundles via Cartier operators in characteristic three