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.

该博士学位关注代数数论中关于模形式和伽罗瓦表示之间的联系的问题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计划的一部分。该项目的特别重点是研究有关 L 函数特殊值的 Bloch-Kato 猜想。这些猜想预测，与伽罗瓦表示相关的解析 L 函数值的 p 整除性恰好对应于与该表示相关的代数对象（即所谓的 Selmer 群）大小的 p 整除性。 theta 升力的 p 积分及其范数可以为布洛赫-加藤猜想的​​更高阶情况提供证据。该博士项目研究了伽罗瓦表示的 Asai 表示与虚二次域上 GL(2) 的自同构表示（Bianchi 模形式） p 的猜想。目标是证明，如果质数 p 除以 Asai 表示的 L 值，则 theta lift（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。然后可以使用与这些形式相关联的伽罗瓦表示来构造 Selmer 群中的元素。这种涉及模形式同余的策略是由 Ribet 在证明 Herbrand 定理的逆过程中首创的，此后被 Wiles 和 Skinner 以及 Urban for Eisenstein 系列用于证明 Iwasawa 主要猜想。该项目与之前关于布洛赫-加藤猜想的​​工作有很大不同，因为它在不知道该表示是否具有“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重量西格尔模块化形式。