Dimensions of Deformation Rings and Automorphy Lifting Theorems

这个研究项目研究代数数论中的问题。数论和自守形式之间的联系是周期函数的推广，是一个非常活跃的研究领域，并导致了数论中长期存在的问题的解决方案。这样的工作有应用程序解决丢番图方程-代数方程与整数系数的整数解决方案寻求-这在广泛的应用程序中发挥作用，从加密协议的纠错。 此外，在代数数论领域开发的方法可以在密码学中有其他令人惊讶的应用。本计画旨在加深对这一重要课题的认识，研究的问题与数论中的经典问题有关，包括Leopoldt猜想和高维类似物。将使用的方法涉及自同构提升定理，这在重要的发展中发挥了至关重要的作用，如费马大定理的证明和塞尔猜想的证明。该项目旨在扩充这些方法并使其更加灵活，这将导致在多项式方程的解及其与自守形式的关系领域中的更多应用。

This research project studies questions in algebraic number theory. The connection between number theory and automorphic forms, which are generalizations of periodic functions, is a very active area of investigation and has led to solutions of longstanding problems in number theory. Such work has applications to solving Diophantine equations -- algebraic equations with integer coefficients for which integer solutions are sought -- which play a role in a wide range of applications from encryption protocols to error correction. Additionally, methods developed in the area of algebraic number theory can have other surprising applications to cryptography. This project aims to deepen knowledge in this important subject.The questions that the project studies are related to classical problems in number theory, including the Leopoldt conjecture and higher dimensional analogs. The methods to be used relate to automorphy lifting theorems, which have played a crucial role in important developments such as the proof of Fermat's Last Theorem and the proof of Serre's conjecture. The project aims to augment these methods and to make them more flexible, which will lead to more applications in the field of solutions of polynomial equations and its relation to automorphic forms.