In the present paper we study Frobenius-projective structures on projective smooth curves in positive characteristic. The notion of Frobenius-projective structures may be regarded as an analogue, in positive characteristic, of the notion of complex projective structures in the classical theory of Riemann surfaces. By means of the notion of Frobenius-projective structures we obtain a relationship between a certain rational function, i.e., a pseudo-coordinate , and a certain collection of data which may be regarded as an analogue, in positive characteristic, of the notion of indigenous bundles in the classical theory of Riemann surfaces, i.e., a Frobenius-indigenous structure . As an application of this relationship, we also prove the existence of certain Frobenius-destabilized locally free coherent sheaves of rank two.

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In the present paper, we study the p-adic Teichmüller theory in the case where p ¼ 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 characterization, we prove that, for every nilpotent ordinary indigenous bundle over a projective smooth curve in characteristic three, there exists a connected finite étale 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 Teichmüller theory, respectively.

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