The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-18 DOI:10.4310/cjm.2024.v12.n2.a2

Ashay Burungale, Matthias Flach

引用次数: 0

Abstract

Let $E/F$ be an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in an imaginary quadratic field $K$. We give a complete proof of the conjecture of Birch and Swinnerton-Dyer for $E/F$, as well as its equivariant refinement formulated by Gross $\href{https://doi.org/10.1007/978-1-4899-6699-5_14}{[39]}$, under the assumption that $L(E/F, 1) \neq 0$ and that $F(E_{tors})/K$ is abelian. We also prove analogous results for CM abelian varieties $A/K$.

查看原文本刊更多论文

Birch 和 Swinnerton-Dyer 对某些具有复乘法的椭圆曲线的猜想

设 $E/F$ 是一条数域 $F$ 上的椭圆曲线，其复数乘以虚二次域 $K$ 中的整数环。在 $L(E/F, 1) \neq 0$ 和 $F(E_{tors})/K$ 是无等边的假设下，我们给出了伯奇和斯温纳顿-戴尔对 $E/F$ 的猜想及其由格罗斯 $\href{https://doi.org/10.1007/978-1-4899-6699-5_14}{[39]}$ 提出的等变细化的完整证明。我们还证明了 CM 无性变项 $A/K$ 的类似结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.