In the present paper, we study the
p-adic Teichmüller 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 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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