Automorphic forms, Galois representations and ramification

项目“自同构形式，伽罗瓦表示和分支”处理占据现代代数数论的一些关键问题。它们涉及对有理的代数扩展的研究，以及它们的一些更难以捉摸的性质，这些性质不能通过直接手段获得。人们可以使用自同构形式来构造有理数的代数扩展，这样就可以更好地控制一些基本性质，人们想知道关于这些扩展的一些基本性质。PI建议通过使用模形式之间的同余来产生具有控制分支的数域的扩展来继续他的工作主题。这种控制与利奥波德猜想等经典猜想有关。他还将继续研究伽罗瓦表示的自同构形式，并证明更多反伽罗瓦问题的情况。研究自同构形式和数论之间的联系是数论中一个非常活跃的领域。这对数论在密码学等实际应用领域的应用具有广泛的意义。PI还希望通过让学生参与研究项目、组织会议和针对广泛受众的演讲来继续传播知识。

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.