The ordinarity of an insotrivial elliptic fibration

TL;DR

This paper investigates the conditions under which an isotrivial elliptic surface over a field of positive characteristic is ordinary, linking the ordinarity of the surface to that of its fibers and certain base covers.

Contribution

It establishes a criterion connecting the ordinarity of an isotrivial elliptic surface with the ordinarity of its fibers and a finite cover of its base, leading to an ordinary reduction theorem.

Findings

An isotrivial elliptic surface is ordinary if its fiber and a finite base cover are ordinary.

The paper provides a criterion for ordinarity based on fiber and base properties.

It derives an ordinary reduction theorem for specific isotrivial elliptic fibrations.

Abstract

In this paper, we study the ordinarity of an isotrivial elliptic surface defined over a field of positive characteristic. If an isotrivial elliptic fibration $\pi:X \to C$ is given, $X$ is ordinary when the common fiber of $\pi$ is ordinary and a certain finite cover of the base $C$ is ordinary. By this result, we may obtain the ordinary reduction theorem for some kinds of isotrivial elliptic fibrations.