EPSRC Reference: 
EP/S031537/1 
Title: 
Moduli of Elliptic Curves and Classical Diophantine Problems 
Principal Investigator: 
Siksek, Professor S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
Standard Research 
Starts: 
01 October 2019 
Ends: 
30 September 2022 
Value (£): 
386,239

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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.

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Summary 

Date Materialised 


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Further Information: 

Organisation Website: 
http://www.warwick.ac.uk 