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Reduction of bielliptic surfaces

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  12 February 2025

Teppei Takamatsu Open the ORCID record for Teppei Takamatsu [Opens in a new window]
Teppei Takamatsu*
Affiliation:
Department of Mathematics (Hakubi Center), Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
*
e-mail: teppeitakamatsu.math@gmail.com
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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 article, 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 does not generally hold for bielliptic surfaces without rational points. In particular, this article completes the study of the Shafarevich conjecture for minimal surfaces of Kodaira dimension $0$. In the second part of this article, we study a Néron model of a bielliptic surface. We establish the potential existence of a Néron model for a bielliptic surface when the residual characteristic is not equal to $2$ or $3$.

Keywords

Bielliptic surfacesShafarevich conjectureNéron model

MSC classification

Primary: 11G35: Varieties over global fields
Secondary: 11G25: Varieties over finite and local fields

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 3 , September 2025 , pp. 856 - 873
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work was supported by JSPS KAKENHI Grant number JP19J22795, JP22J00962, JP22KJ1780

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