论文标题
关于椭圆形曲线的模块化，元素$ \ mathbb {z} _p $ - 某些真实二次字段的扩展
On the modularity of elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension of some real quadratic fields
论文作者
论文摘要
椭圆曲线的模块化总是吸引人数为理论家。最近，索恩（Thorne）证明了一个奇妙的结果，对于$ p $ $ p $ cyclotomic的扩展为$ \ mathbb {q} $的每个椭圆曲线都是模块化的。该方法是使用一些自动形成提升定理，并在伊瓦泽理论上在某些特定的椭圆曲线上研究非CUSP点，以用于椭圆曲线。由于已证明了椭圆曲线的模块化，因此可以询问是否可以用真正的二次字段$ k $替换$ \ mathbb {q} $。遵循Thorne的想法，我们首先给出一些假设，并在$ \ Mathbb {z} _p $ - 某些实际二次域字段的extension上证明椭圆曲线的模块化。
The modularity of elliptic curves always intrigues number theorists. Recently, Thorne had proved a marvelous result that for a prime $ p $, every elliptic curve defined over a $ p $-cyclotomic extension of $ \mathbb{Q} $ is modular. The method is to use some automorphy lifting theorems and study non-cusp points on some specific elliptic curves by Iwasawa theory for elliptic curves. Since the modularity of elliptic curves over real quadratic was proved, one may ask whether it is possible to replace $ \mathbb{Q} $ with a real quadratic field $ K $. Following Thorne's idea, we give some assumptions first and prove the modularity of elliptic curves over the $\mathbb{Z}_p$-extension of some real quadratic fields.
