Congruences of automorphic forms and Galois representations

这个博士学位涉及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问题在算术问题中的应用。建立自守形式和伽罗瓦表示之间的精确联系是著名的朗兰兹纲领的一部分。该项目的特别重点将是研究关于L函数特殊值的布洛赫-加藤定理。这些理论预言了与伽罗瓦表示相关联的解析L-函数的值的p-整除性恰好对应于与该表示相关联的代数对象的大小的p-整除性，即所谓的塞尔默群。关于θ提升及其范数的p-整性的结果使得有可能为布洛赫-加藤猜想的更高秩情况提供证据。博士项目研究了这个猜想的Asai表示的伽罗瓦表示相关的一个自守表示（比安奇模形式）p GL（2）在一个虚二次域。目标是证明如果素数p整除Asai表示的L值，则theta提升（与p相关的西格尔模形式）和其他“稳定”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塞尔默群中的元素。这一策略涉及同余的模块形式是开创性的里贝特在他的证明匡威Herbrand定理，并已被使用，其中包括怀尔斯和斯金纳和城市的爱森斯坦系列的证明岩泽主要progratures。该项目是显着不同的布洛赫-加藤结构，因为它提供了证据的背景下，它是不知道是否表示是“动机”以前的工作。在其他技术中，它需要扩展拉回法证明同余低重量西格尔模块化形式。

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.