Modular varieties, generalized Fermat equations, and special functions

从有记录的历史开始，数论就已经存在了，它的动机是研究数字和方程式。也许是因为它在数学中的基础作用，数论继续是一个活跃的研究领域，具有广泛的适用性和应用相关性。 这项研究计划旨在研究和回答数论中一些基本的公开问题，这些问题涉及一类自然的丢番图方程，称为模簇，以期开发出更强大的方法来解决费马-加泰罗尼亚猜想，费马最后定理的自然推广。 该方法是基础科学和问题驱动的，但也将涉及伽罗瓦表示、模符号和Frey阿贝尔变化构造领域的新方法和理论的发展。 在应用的相关性方面，所提出的研究加强了我们关于椭圆和超椭圆曲线、数域和函数域的基础知识，以及表示理论的显式方面。 从该提案的基础科学部分获得的专业知识还将用于研究同态加密的应用驱动问题及其对量子算法的抵抗，这与大数据应用中的隐私保护相关。

Number theory, which is motivated by the study of numbers and equations, has existed since the dawn of recorded history. Perhaps because of its fundamental role in mathematics, number theory continues to be an active area of research, with a wide applicability and relevance to applications. This research program aims to study and answer some fundamental open questions in number theory concerning a natural class of diophantine equations known as modular varieties, with a view towards developing stronger methods for tackling the Fermat-Catalan conjecture, a natural generalization of Fermat's Last Theorem. The approach is fundamental science and problem motivated but will also involve the development of new methods and theory in the areas of Galois representations, modular symbols, and Frey abelian variety constructions. In terms of relevance to applications, the proposed research enhances our foundational knowledge concerning elliptic and hyperelliptic curves, number and function fields, as well as explicit aspects of representation theory. The expertise gained from the fundamental science component of this proposal will also be used to study the application motivated problem of homomorphic encryption and its resistance to quantum algorithms, which is relevant to maintaining privacy in big data applications.