GENUS CURVES WITH BAD REDUCTION AT ONE ODD PRIME

Pub Date : 2023-11-29 DOI:10.1017/nmj.2023.35

ANDRZEJ DĄBROWSKI, MOHAMMAD SADEK

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Abstract

The problem of classifying elliptic curves over

$\mathbb Q$

with a given discriminant has received much attention. The analogous problem for genus

$2$

curves has only been tackled when the absolute discriminant is a power of

$2$

. In this article, we classify genus

$2$

curves C defined over

${\mathbb Q}$

with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many

$1$

-parameter families of genus

$2$

curves. In particular, we provide genus

$2$

analogues to Neumann–Setzer families of elliptic curves over the rationals.

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在一个奇素数处约化不良的属曲线

用给定的判别式对$\mathbb Q$上的椭圆曲线进行分类的问题受到了广泛的关注。对于$2$曲线的类似问题，只有在绝对判别式是$2$的幂次时才得到解决。在这篇文章中，我们对定义在${\mathbb Q}$上的$2$曲线C进行了分类，这些曲线C至少有两个有理Weierstrass点，其绝对判别式是奇素数。事实上，我们证明了这样的曲线C必须同构于有限多个$1$参数族的$2$曲线中的一个的专门化。特别地，我们提供了在有理数上的椭圆曲线的Neumann-Setzer族的属$2类似物。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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