Explicit methods for algebraic automorphic forms

朗兰兹计划是现代数学中最雄心勃勃的研究计划之一。该程序是一个巨大的猜想网络，将分析、代数几何、表示理论和数论联系在一起。该程序的核心是自同构形式(分析对象)和伽罗瓦表示(几何对象)。这两个集合是由一系列猜想联系在一起的，这些猜想的精度根据我们的理解而有所不同。该方案值得我们关注，因为它具有深远的影响。这个项目将开发代数自同构形的显式方法。通过提供可以使用的自同构的具体例子，它将帮助我们更好地理解该领域的许多猜想。私家侦探对摩登·朗兰兹特别感兴趣，因为他在这起案件中与GL(2)集团取得了一些成功。

The Langlands Programme is one of the most ambitious research programmes in modern mathematics. The programme is a vast web of conjectures which connect analysis, algebraic geometry, representation theory and number theory. At the heart of this programme are automorphic forms (analytic objects) and Galois representations (geometric objects). The two sets are related by a series of conjectures whose level of precision varies according to our understanding. The programme warrant our attention as it has far reaching consequences. This project will develop explicit methods for algebraic automorphic forms. By providing concrete examples of automorphic to work with, it will help improve our understanding of many of conjectures in this field. The PI is particularly interested in the mod p Langlands, as he has had some success in this case with the group GL(2).