Congruences of automorphic forms and Galois representations

自守形式和伽罗瓦表示的同余

• 批准号: 2745671
• 金额: --
• 依托单位: University of Sheffield
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2745671
• 关键词: Congruences; automorphic; forms; Galois; representations

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.

该博士学位涉及代数数理论中有关模块化形式与Galois表示之间的联系以及这些与算术问题的应用之间的联系的问题。著名的Langlands计划的一部分是建立自动形式和Galois代表之间的精确联系。该项目的特殊重点是对Loch-kato的特殊值进行bloch-kato的猜想。这些猜想预测，与加洛伊斯表示相关的分析l功能值的p划分完全与与代表相关的代数对象的大小的p-划分性，所谓的selmer群体，所谓的selmer群体。对theta提升的p-积分性及其规范性的证明使得有可能证明是一个证明了一个构成的构成，以证明是相关的，因为它可以证明是相关的。博士项目研究了该猜想对与想象中二次场上的GL（2）相关的GLOIS表示的ASAI表示形式。目的是证明，如果prime p划分了ASAI表示的L值，则存在Theta升降机（与P相关的Siegel模块化形式）与其他“稳定” Siegel模块化形式之间的一致性。然后，与这些形式关联的GALOIS表示可以用于在Selmer组中构造元素。 Ribet在他的Converse证明了Herbrand定理的证明中率先提出了涉及模块化形式的一致性的策略，此后，在伊瓦苏瓦主要猜想的证明中，埃森斯坦（Eisenstein）系列的这种策略已被埃森斯坦（Eisenstein）系列使用。该项目与以前关于Bloch-Kato猜想的工作有很大不同，因为它在不知道该代表是否是“动机”的情况下提供了证据。在其他技术中，它需要将回调方法扩展到低重量siegel模块化形式。