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。该计划是一个庞大的网络的acetutures连接分析，代数几何，表示论和数论。这个程序的核心是自守形式（解析对象）和伽罗瓦表示（几何对象）。这两个集合由一系列的结构联系起来，其精确程度根据我们的理解而变化。该方案值得我们注意，因为它具有深远的影响。本计画将发展代数自守形式的显式方法。通过提供自守的具体例子，它将有助于提高我们对该领域许多知识的理解。PI对mod p Langlands特别感兴趣，因为他在GL（2）群的情况下取得了一些成功。

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).