Arithmetic Moduli Spaces and Gauge Theory

This proposal is concerned with the theory of Diophantine equations, that is, the study of rational or integral solutions to algebraic equations. This is one of the oldest subjects in mathematics, going back possibly to the ancient Babylonians and systematised by Diophantus of Alexandria around the 3rd century. Nevertheless, it is still the source of some of the most difficult problems and wide-ranging programmes in mathematics, such as Fermat's Last Theorem or the conjectures of Birch and Swinnerton-Dyer. In spite of much progress over the last 100 years or so using the modern methods of arithmetic geometry, the major problems remain unsolved, especially when it comes to algorithmic methods that can find solutions to equations on a computer. (This is furthermore hampered by certain impossibility theorems of mathematical logic.) This research proposes to apply new ideas inspired by high energy physics to the study of Diophantine equations in two variables based on an analogy between the solution space to Euler-Lagrange equations in physics and the non-Archimedean geometry of 'arithmetic gauge fields' constructed by the PI. The main goal is to generalise to higher degree the methodology surrounding the conjecture of Birch and Swinnerton-Dyer, which is concerned with equations of degree 3. Eventually, this research should lead to substantial progress on the problem of devising a computer algorithm that will find all rational solutions to equations of degree at least 4 and two unknowns.