Abstract
This thesis studies integer solutions to polynomial equations, using modern algebraic geometry. Our focus lies on understanding the arithmetic properties of geometric objects defined by these equations, which in turn sheds light on the solutions to the original polynomial equations.To begin with we study two different families of cubic polynomials, which can be seen as generalisations of the sum of three cubes conjecture. In particular, we consider whether the existence of local solutions is not sufficient for the existence of global solutions, i.e. do any of these equations fail the integral Hasse principle? To do so we use the integral Brauer--Manin obstruction. In the first case we show that the integral Brauer--Manin obstruction does not yield any failure of the integral Hasse principle for nearly all these equations.
In studying the second family of cubic equations (which is joint work with Julian Lyczak and Vladimir Mitankin) we show that the integral Brauer--Manin obstruction can give failures of the integral Hasse principle, providing the first examples of such equations failing the integral Hasse principle. Furthermore, we also quantify the frequency of these failures.
We proceed by studying a family of polynomial equations which admit a special geometric structure, namely a fibration of torsors under norm 1 tori. Here we give sufficient conditions for the existence of integer solutions using the descent-fibration method.
Finally, we move on to studying singular del Pezzo surfaces over finite fields in an attempt to provide an analogue of a conjecture of Lang. Explicitly, we want to know if there is a smooth point on these surfaces. We show that away from characteristic 2, this is always true, however there are counterexamples in the case of characteristic 2.
| Date of Award | 22 Jan 2025 |
|---|---|
| Original language | English |
| Awarding Institution |
|
| Supervisor | Daniel Loughran (Supervisor) |