Deformation in p-adic families of Bianchi modular forms

模形式的理论及其推广，自同构形式，是现代数论的核心。研究这些理论的一种方法是通过特征曲线构造，这些曲线的几何性质对应于算术结果。我将重点关注特征曲线的平滑性，在非规则权重下，以及在除GL_2以外的有理数组上。在二次虚场上的GL_2中，Loeffler和Zerbes的“关于模阿贝尔曲面的Birch-Swinnerton-Dyer猜想”表明，光滑性意味着（在一些附加假设下）Birch-Swinnerton-Dyer猜想的特殊情况。对于GL_2 / Q，对于权重大于1的点，平滑性是众所周知的。Bellaiche和Dimitrov最近的作品《经典权重1点的特征曲线》（On the Eigencurve at classic weight 1）也证实了这一点。Calegari和Mazur在“任意数域上的近普通伽罗瓦变形”中探讨了GL_2在除Q以外的数域上的这些问题，但仍有许多问题没有得到解答。此外，这些作品通过附加在模形式上的伽罗瓦表示的变形来解决问题，而我的目标是更直接地处理模形式和模符号。这种方法的另一个好处是计算结果的可能性-我已经开发了一种算法来数值测试欧几里得域上尖头形式的平滑性，我希望将其扩展到更一般的二次虚域。我还希望扩展这些想法，为平滑性提供一个通用标准。

The theory of modular forms and their generalisation, automorphic forms, is central to modern number theory. One way to study these theories is via the eigencurve construction, where geometric properties of these curve correspond to arithmetic results. I will focus on smoothness of eigencurves, at non-regular weights and for groups other than GL_2 over the rationals. In the case of GL_2 over quadratic imaginary fields, work by Loeffler and Zerbes, "On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces", shows smoothness would imply (under some additional hypotheses) special cases of the Birch-Swinnerton-Dyer conjecture.For GL_2 / Q, smoothness is well known for points of weight greater than one. Recent work by Bellaiche and Dimitrov, "On the Eigencurve at classical weight 1 points", establishes this at weight one as well. Calegari and Mazur have explored these questions for GL_2 over number fields other than Q in "Nearly Ordinary Galois Deformations over Arbitrary Number Fields", but much remains unanswered. Moreover, these works approach the problem via deformations of the Galois representations attached to modular forms, whereas I aim to work more directly with modular forms and modular symbols. An additional benefit of this method is the possibility for computational results - I have developed an algorithm to numerically test for smoothness of cusp forms over Euclidean domains, and I hope to extend this to more general quadratic imaginary fields. I also hope to extend these ideas to provide a general criterion for smoothness.