Automorphic forms, Galois representations and ramification

项目 “自守形式，伽罗瓦表示和分歧”涉及占据现代代数数论的一些关键问题。它们涉及到有理数的代数扩展的研究，以及它们的一些更难以捉摸的性质，这些性质无法通过直接手段获得。 人们使用自守形式来构造有理数的代数扩张，这使得人们对这些扩张的一些基本性质有了更大的控制。 PI建议继续这一主题，他的工作，使用同余之间的模块形式，以产生扩展的数域与控制的分歧。这样的控制是与经典的猜想，如利奥波德猜想。他还将继续研究附加自守形式的伽罗瓦表示，并证明更多的情况下，逆伽罗瓦问题。工作之间的连接自守形式和数论是一个非常活跃的领域数论。这具有广泛的影响，可能也是有用的数论应用领域，如密码学是实际使用。PI还希望继续disemminent知识，通过让学生参与研究项目，通过组织会议，并给予针对广泛的受众对他的主题的讲座。

The project ``Automorphic forms, Galois representations and ramification'' deals with some of the key issues that occupy modern algebraic number theory. They relate to the study of algebraic extensions of the rationals, and some of their more elusive properties that are not accessible by direct means. One uses automorphic forms to construct algebraic extensions of the rationals, which gives greater control on some of the basic properties that one would like to know about such extensions. The PI proposes to continue this theme of his work by using congruences between modular forms to produce extensions of number fields with controlled ramification. Such a control is related to classical conjectures like the Leopoldt conjecture. He will also continue the study of attaching automorphic forms to Galois representations, and proving more cases of the inverse Galois problem.Work on the connection between automorphic forms and number theory is a very active area of number theory. This has broad implications which might also be useful to applications of number theory in areas like cryptography that are of practical use.The PI also expects to continue to disemminate knowledge by involving students in research projects, by organising conferences, and giving talks aimed at broad audiences about his subject.