Moduli of Elliptic Curves and Classical Diophantine Problems

如果我们寻求整数或分数的解，我们称之为丢芬图。这些方程式是以亚历山大的丢番图命名的，他可能生活在公元三世纪左右。然而，这个主题要古老得多；例如，一块大约4000年前的巴比伦泥板列出了现在被称为毕达哥拉斯方程的小整数解。丢番图方程的主题在17世纪法国法学家和业余数学家皮埃尔·德·费马的复兴和普及。特别是，费马大定理是一个公开的丢芬图问题，它俘获了公众250多年的想象力，最终由安德鲁·怀尔斯（Andrew Wiles）在1994年解决。虽然费马大定理的陈述和许多其他丢芬图问题可以被任何受过教育的人理解，但这门学科是当代数学中最深奥的学科之一，它与代数几何、分析和表示理论等其他数学学科建立了深刻的联系。拟议的研究包括两个主题。第一种是模曲线，模曲线本质上是丢番图方程，它的解将其他丢番图对象分类，称为椭圆曲线。模曲线在现代数论中起着至关重要的作用，是解决若干难以解决的问题的关键。在这个项目中，我们开发了理论和计算工具来研究模曲线的算法。第二个主题是关于经典丢芬图问题的某些家族，其中贝克的理论为解决方案提供了边界，但搜索区域如此之大，甚至超过了最强大的计算机集群的计算能力。我们将开发利用格理论筛选搜索区域的新技术。

An equation is called Diophantine if we seek solutions that are either whole or fractional numbers. These equations are named after Diophantus of Alexandria who probably lived around the third century AD. However, the subject is far older; for example a Babylonian clay tablet around 4000 years old lists small whole number solutions of what is now known as the Pythagorean equation. The subject of Diophantine equations was revived and popularised by 17th century French jurist and amateur mathematician Pierre de Fermat. In particular, Fermat's Last Theorem was an open Diophantine problem that captured public imagination for over 250 years and was finally settled by Andrew Wiles in 1994. Whilst the statement of Fermat's Last Theorem and many other Diophantine problems can be understood by any educated person, the discipline is one of the deepest in contemporary mathematics, and builds on profound connections with other mathematical disciplines such as algebraic geometry, analysis and representations theory.The proposed research comprises of two themes. The first is concerned with modular curves, which in essence are Diophantine equations whose solutions classify certain other kinds of Diophantine objects called elliptic curves. Modular curves play a crucial role in modern number theory, and are the key to several difficult unresolved problems. In this project we develop theoretical and computational tools for studying the arithmetic of modular curves.The second theme is concerned with certain families of classical Diophantine problems where Baker's theory provides bounds for the solutions but the search regions are so enormous that they are beyond the computational capabilities of even the most powerful computer clusters. We will develop new techniques for sifting search regions using the theory of lattices.