Reduction of bielliptic surfaces

TL;DR

This paper investigates the structure and Néron models of bielliptic surfaces, proving the Shafarevich conjecture for those with rational points over certain fields, and demonstrating its failure without rational points.

Contribution

It provides a detailed study of bielliptic surfaces, proving the Shafarevich conjecture in specific cases and exploring the existence of Néron models in various characteristics.

Findings

Shafarevich conjecture holds for bielliptic surfaces with rational points in characteristic ≠ 2,3

The conjecture generally fails for bielliptic surfaces without rational points

Potential existence of Néron models when residual characteristic ≠ 2,3

Abstract

A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this paper, we study the structure of bielliptic surfaces over a field of characteristic different from $2$ and $3$, in order to prove the Shafarevich conjecture for bielliptic surfaces with rational points. Furthermore, we demonstrate that the Shafarevich conjecture generally fails for bielliptic surfaces without rational points. In particular, this paper completes the study of the Shafarevich conjecture for minimal surfaces of Kodaira dimension $0$. In the second part of this paper, we study a N\'{e}ron model of a bielliptic surface. We establish the potential existence of a N\'{e}ron model for a bielliptic surface when the residual characteristic is not equal to $2$ or $3$.