Diophantine equations and local-global principles: into the wild (renewal)

This project sits within the realm of Diophantine equations, an ancient area of mathematics containing problems that have remained unsolved for thousands of years. Mathematicians working in this field study the integer (i.e. whole number) solutions of polynomial equations. Given such an equation, the first question one might ask is whether any integer or rational solutions exist. In fact, this can be a very hard problem: even with the most powerful computers in the world, we cannot check all the infinitely many possibilities to see whether they are solutions. In 1900, Hilbert challenged mathematicians to come up with an algorithm that can determine whether a polynomial equation has an integer solution. Seventy years later, building on work of Robinson, Davis and Putnam, Matiyasevich showed that no such general algorithm exists! Nevertheless, the study of these equations remains a thriving area of current research and they underlie cryptographic schemes protecting our data in many aspects of modern life, e.g. online shopping. Modern methods for tackling Diophantine equations proceed via the local-global method. One first checks whether the equation has an integer solution everywhere locally. This is a finite computation, thanks to the Lang—Weil bounds. The challenging part is deciding whether these local solutions patch together to form a global integer solution. For some types of equations, such as quadratic forms, this always works. This is what it means to say that the Hasse principle (the primary example of a local-global principle) holds for quadratic forms. But for equations of higher degree, such as cubic equations, the Hasse principle can fail. Understanding why and how often local-global principles fail is key to unravelling the mysteries of Diophantine equations, and is the focus of this research project. A common explanation for the failure of local-global principles comes from the so-called Brauer—Manin obstruction. Skorobogatov has conjectured that the Brauer—Manin obstruction explains all failures of the Hasse principle for equations defining K3 surfaces. These surfaces are of interest for several reasons: they sit at the boundary of what is known about Diophantine equations, have been used as a testing ground for important conjectures (e.g. Deligne’s proof of the Weil conjectures), and also crop up in mirror symmetry and string theory. The first strand of my project concerns Brauer—Manin obstructions on K3 surfaces. In certain cases of interest, I will calculate the relevant Brauer groups and from there compute Brauer—Manin obstructions by evaluating Brauer group elements at collections of local points and investigating how these evaluations vary as one changes the local points. The second strand of my project takes a statistical approach and studies whole families of mathematical objects. For example, I will study a certain family of Diophantine equations that define geometric objects called elliptic curves and aim to determine how often their solution sets are finite as opposed to infinite. Equations defining elliptic curves rightfully command a great deal of attention. They are noted for the rich structure of their sets of solutions in which one can build new solutions from old, their connection to ancient problems (such as the still unsolved Congruent Number Problem that dates back to at least the 10th century), their crucial role in Wiles’ famous proof of Fermat’s Last Theorem, and their use in elliptic curve cryptography.