Congruences of automorphic forms and Galois representations

本博士关注代数数论中关于模形式和伽罗瓦表示之间的联系以及这些在算术问题中的应用的问题。建立自同构形式和伽罗瓦表示法之间的精确联系是著名的朗兰兹纲领的一部分。这个项目的重点是研究关于l函数的特殊值的Bloch-Kato猜想。这些猜想预测，与伽罗瓦表示相关联的解析l函数值的p可分性与与该表示相关联的代数对象的大小的p可分性完全对应，即所谓的Selmer群。关于提升及其范数的p-完整性的结果使它有可能为更高秩的Bloch-Kato猜想提供证据。博士项目研究了虚二次域上与GL(2)的自同构表示（Bianchi模形式）p相关的伽罗瓦表示的Asai表示的这个猜想。我们的目标是证明如果一个素数p能除Asai表示的l值，那么在一个提升（与p相关的西格尔模形式）和其他“稳定的”西格尔模形式之间存在同余。与这些形式相关联的伽罗瓦表示可以用来构造Selmer组中的元素。这种涉及模形式同余的策略是由Ribet在他对Herbrand定理的逆证明中首创的，并且后来被Wiles， Skinner和Urban等人用于爱森斯坦级数在Iwasawa主要猜想的证明中。该项目与之前关于Bloch-Kato猜想的工作有很大不同，因为它在不知道表征是否“动机”的情况下提供了证据。在其他技术中，它需要将证明同余的回拉方法扩展到低权重西格尔模形式。

This PhD concerns questions in algebraic number theory about connections between modular forms and Galois representations and applications of these to arithmetic problems. Establishing the precise links between automorphic forms and Galois representations is part of the famous Langlands programme. The particular focus of the project would be to work on the Bloch-Kato conjectures on special values of L-functions. These conjectures predict that the p-divisibility of a value of the analytic L-function associated to a Galois representation corresponds exactly to the p-divisibility of the size of an algebraic object associated to the representation, the so-called Selmer group.Results on the p-integrality of theta lifts and their norm make it possible to provide evidence for a higher rank case of the Bloch-Kato conjecture. The PhD project studies this conjecture for the Asai representation of the Galois representation associated to an automorphic representation (Bianchi modular form) p for GL(2) over an imaginary quadratic field. The goal is to prove that if a prime p divides the L-value for the Asai representation then there exists a congruence between a theta lift (a Siegel modular form associated to p) and other "stable" Siegel modular forms. The Galois representations associated to these forms can then be used to construct elements in the Selmer group. This strategy involving congruences of modular forms was pioneered by Ribet in his proof of the converse of Herbrand's theorem and has since been used by, amongst others, Wiles and Skinner and Urban for Eisenstein series in the proof of Iwasawa main conjectures. The project is significantly different to previous work on the Bloch-Kato conjectures as it provides evidence in a context where it is not known whether the representation is "motivic". Amongst other techniques it requires extending the pullback method of proving congruences to low weight Siegel modular forms.