论文标题
$ \ mathbb {c} $上的模块化意味着$ \ mathbb {q} $上的模块化
Modularity over $\mathbb{C}$ implies modularity over $\mathbb{Q}$
论文作者
论文摘要
我们给出了Mazur的证明，说明，对于$ \ Mathbb {q} $的椭圆曲线，如果它承认从$ x（n）$中定义的$ x（n）$ $ \ mathbb {c} $定义的非固定映射，则对于某些$ n $，它也承认了从$ x_0（MOSPAIND）$ x_0（MOS）（可能是$ x_0（MO）的$ NOMPLIASS（MO）（MOS），以获取$ X_0（MM）（MOS），以供某些$ x_0（MM）（MOS），以获取ress ocy for locess $ nmundice（Mor）（m）$ n $ $ \ mathbb {q} $。我们还简要讨论了Khare的两个关于通过非综合模块化曲线的椭圆曲线均匀化的开放问题。 该说明性票据基于2022年3月在印度浦那的Bhaskaracharya Pratishthana在线举行并组织的关于“模块化和广义费马特方程式”的第二学期计划中的作者的说明性演讲。
We give an account of Mazur's proof that, for an elliptic curve over $\mathbb{Q}$, if it admits a nonconstant mapping from $X(N)$ defined over the complex numbers $\mathbb{C}$, for some $N$, then it also admits a nonconstant mapping from $X_0(M)$ for some (possibly different) $M$ defined over the rational numbers $\mathbb{Q}$. We also briefly discuss two open questions of Khare concerning uniformisations of elliptic curves by noncongruence modular curves. This expository note is based on the author's expository talk in March 2022 during the 2nd Trimester Program on "Modularity and the Generalised Fermat Equation" held online and organised by Bhaskaracharya Pratishthana in Pune, India.
