Moduli of Elliptic Curves and Classical Diophantine Problems

一个方程叫做丢番图，如果我们寻求整数或分数的解。这些方程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，这个问题要古老得多;例如，一个大约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。丢番图方程的主题是由17世纪世纪法国法学家和业余数学家皮埃尔·德·费马复兴和普及的。特别是，费马大定理是一个开放的丢番图问题，吸引了公众超过250年的想象力，并最终在1994年由安德鲁·怀尔斯解决。虽然费马大定理和许多其他丢番图问题的陈述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的学科之一，并建立在与其他数学学科如代数几何，分析和表示理论的深刻联系之上。第一个是关于模曲线，这在本质上是丢番图方程的解决方案分类某些其他种类的丢番图对象称为椭圆曲线。模曲线在现代数论中起着至关重要的作用，并且是一些尚未解决的难题的关键。在这个项目中，我们开发的理论和计算工具，研究算法的模块curves. Second主题是关注某些家庭的经典丢番图问题，贝克的理论提供了边界的解决方案，但搜索区域是如此之大，他们超出了计算能力，即使是最强大的计算机集群。我们将开发新的技术，筛选搜索区域使用的理论格。

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.