Brauer groups of K3 surfaces

Lead Research Organisation: Imperial College London
Department Name: Dept of Mathematics

Abstract

The local-to-global principle for rational points on K3 surfaces over number fields is a difficult problem where very little progress has been achieved so far. This question involves understanding solutions of, among others, quartic equations in three variables over the rationals. Since the local-to-global behaviour of rational points on K3 surfaces is expected to be completely controlled by the Brauer-Manin obstruction, the computation of the Brauer group of K3 surfaces is an important problem too. A recent conjecture of Várilly-Alvarado predicts that only finitely many finite abelian groups can appear as transcendental Brauer groups of K3 surfaces defined over a given number field. The aim of this research is to use various recent achievements in the Tate and Mumford-Tate conjectures, André-Oort conjecture and so on, to advance in the direction of the conjecture of Várilly-Alvarado. A novel methodology involves applying isogenies of K3 surfaces to link the Brauer groups of related K3 surfaces. The similar question regarding abelian varieties is also an interesting open question. The arithmetic theory of K3 surfaces is fast becoming an active field of research due to potential impact on classical difficult questions of number theory and arithmetic geometry.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509486/1 01/10/2016 31/03/2022
1832319 Studentship EP/N509486/1 01/10/2016 31/03/2020 Domenico Valloni
 
Description During the work funded through this award I studied particular objects in mathematic called K3 surfaces with complex multiplication. My results provide a general theory for tackling many interesting questions, especially regarding their Brauer group, fields of definition and Neron-Severi groups. These arithmetic invariants are very important when it comes to practical applications, e.g. computing the Brauer-Manin obstruction in the quest for rational points, and up until now the experts were mostly employing different strategies for different cases. In this sense, the work carried out during this award make it possible to consider all the cases at once, thus simplifying ideas and computations a great deal.
Exploitation Route My work will be useful to experts working on K3 surfaces with complex multiplication. The algorithm I provide to predict the Brauer group of a given K3 surface with CM is particularly useful to whoever is interested in computing the Brauer-Manin obstruction.
Sectors Other

URL https://arxiv.org/abs/1804.08763