J.Bella\"iche and M.Dimitrov have shown that the $p$-adic eigencurve is smooth but not etale over the weight space at $p$-regular classical points having multiplication by a real quadratic field in which $p$ splits. The main purpose of this paper is to show that there exists an isomorphism between a subring of the local ring of the eigencurve at these points and an universal ring representing a pseudo-deformation problem and we show also by using pseudo-deformation and Iwasawa theory that the ramification index is exactly $2$ at certain RM points.

中文翻译：

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经典 RM 点的特征曲线的分支

J.Bella\"iche 和 M.Dimitrov 已经证明 $p$-adic 特征曲线在 $p$-常规经典点的权重空间上是平滑的，但不完全是通过与 $p$ 分裂的实二次域相乘的. 本文的主要目的是证明在这些点的特征曲线的局部环的子环和代表伪变形问题的通用环之间存在同构，我们还通过使用伪变形和岩泽理论来证明在某些 RM 点上，分支指数正好是 2 美元。