具有有理j不变量的非cm椭圆曲线素数域上的等同性度

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory Pub Date : 2025-06-18 DOI:10.1016/j.jnt.2025.05.007

Ivan Novak

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引用次数: 0

摘要

设E是一条有理j不变量的非cm椭圆曲线。Mazur和Kenku确定了E可能具有的所有可能的有理循环同生度。推广这一理论的一种可能的方法是，将定义在某个固定度为d>；1的数域上的所有可能的循环同生度进行分类。我们确定了p次数域上具有有理j不变量的非cm椭圆曲线的所有可能的循环等同源度，其中p是奇素数。Vukorepa对二次型数域进行了类似的分类，本文完成了二次型数域的素数度的分类。为了证明这一结果，我们利用了有理椭圆曲线伽罗瓦像上的已知结果。特别是，在解决立方情况时，我们使用了基于Rouse和Zureick-Brown的2进伽罗瓦表示的图像数据库。

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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Degrees of isogenies over prime degree number fields of non-CM elliptic curves with rational j-invariant

Let E be a non-CM elliptic curve with rational j-invariant. Mazur and Kenku determined all possible degrees of rational cyclic isogenies that E could have. One possible way of generalizing this would be to classify all possible degrees of cyclic isogenies defined over number fields of some fixed degree

d > 1

We determine all possible degrees of cyclic isogenies of non-CM elliptic curves with rational j-invariant over number fields of degree p, where p is an odd prime. The analogous classification for quadratic number fields was done by Vukorepa, and this paper completes the classification in case when the degree of the number field is prime.

To prove the result, we make use of known results on Galois images of rational elliptic curves. In particular, when solving the cubic case, we use the database of images of 2-adic Galois representations which is due to Rouse and Zureick-Brown.

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来源期刊

Journal of Number Theory 数学-数学

CiteScore

1.30

自引率

14.30%

发文量

122

审稿时长

16 weeks

期刊介绍： The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field. The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory. Starting in May 2019, JNT will have a new format with 3 sections: JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access. JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions. Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.