Moduli of Elliptic Curves and Classical Diophantine Problems

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.

The proposed research will lead to powerful new methods and techniques that will become invaluable to researchers working on Galois representations of elliptic curves, arithmetic of modular curves, or more broadly the arithmetic of curves, and also to computational number theorists, and the Diophantine approximation community. These will be the primary beneficiaries.

Diophantine equations have always been a tool for popularising mathematics, and the particularly results arising from the envisaged work on classical Diophantine problems will have the potential to capture public imagination, and raise awareness of front-line mathematical research.