The wild Brauer-Manin obstruction on K3 surfaces

Pagano, M. (2024)

Doctoral Thesis

In this thesis, rational points on K3 surfaces are studied. In the first part of Chapter 1 the Brauer group and the the Brauer-Manin obstruction are introduced. The second part of Chapter 1 is devoted to the introduction of the refined Swan conductor and the link between the latter and the Brauer-Manin set. The aim of Chapter 2 is to give the first example of a K3 surface over a number field (in fact, over the rational numbers) for which a prime of good reduction is involved in the Brauer-Manin obstruction; this gives a negative answer to a question originally asked by Swinnerton-Dyer. In Chapter 3 the role of primes of good reduction in the Brauer-Manin obstruction is discussed, with a focus on K3 surfaces. Finally, in the last chapter, after introducing some techniques that can be used to compute the refined Swan conductor, several examples of Brauer-Manin obstruction on K3 surfaces are provided, with a focus on Kummer K3 surfaces.

• Algebraic geometry
• Arithmetic geometry
• Number theory
• Brauer groups
• Brauer-Manin obstruction
• K3 surfaces

• All authors: Pagano, M.
• Supervisor: Luijk, R.M. van
• Co-supervisor: Bright, M.J.
• Committee: Derks, G.L.A.; Fiocco, M.; Skorobogatov, A.; Ieronymou, E.; Newton, R.
• Qualification: Doctor (dr.)
• Awarding Institution: Mathematical Institute (MI), Faculty of Science, Leiden University
• Date: 2024-07-04