Nilpotent admissible indigenous bundles via Cartier operators in characteristic three

Abstract

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.

On the supersingular divisors of nilpotent admissible indigenous bundles

“…Now let us recall that, in [2], L. R. A. Finotti studied nilpotent ordinary indigenous bundles on hyperbolic curves of type ð2; 0Þ [cf. also [5], Remark 6.1.2]. Moreover, in [1], I. I. Bouw and S. Wewers studied nilpotent ordinary indigenous bundles on hyperbolic curves of type ð0; 4Þ [cf.…”

“…In [5], the author of the present paper gave a characterization of the supersingular divisors of nilpotent admissible/ordinary indigenous bundles in the case where ðr; pÞ ¼ ð0; 3Þ, i.e., on projective hyperbolic curves of characteristic three. The characterization of [5] asserts that if ðr; pÞ ¼ ð0; 3Þ, then it holds that a given e¤ective divisor on X coincides with the supersingular divisor of a nilpotent admissible indigenous bundle on X if and only if the divisor is reduced and may be obtained by forming the zero locus of a Cartier eigenform [cf. [5], Definition A.8, (ii)] associated to a square-trivialized invertible sheaf [cf.…”

“…The characterization of [5] asserts that if ðr; pÞ ¼ ð0; 3Þ, then it holds that a given e¤ective divisor on X coincides with the supersingular divisor of a nilpotent admissible indigenous bundle on X if and only if the divisor is reduced and may be obtained by forming the zero locus of a Cartier eigenform [cf. [5], Definition A.8, (ii)] associated to a square-trivialized invertible sheaf [cf. [5], Definition A.3] on X [cf.…”

“…[5], Definition A.8, (ii)] associated to a square-trivialized invertible sheaf [cf. [5], Definition A.3] on X [cf. [5], Theorem B].…”

“…[5], Definition A.3] on X [cf. [5], Theorem B]. Moreover, in this case, it holds that the nilpotent admissible indigenous bundle on X is ordinary if and only if either the underlying invertible sheaf of the square-trivialized invertible sheaf is trivial, and the Jacobian variety of X is ordinary, or the underlying invertible sheaf of the square-trivialized invertible sheaf is nontrivial [i.e., of order two], and the Prym variety associated to the underlying invertible sheaf is ordinary [cf.…”