Counting points on K3 surfaces and other arithmetic-geometric objects

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Counting points on K3 surfaces and other arithmetic-geometric objects

Type: Doctoral Thesis
Title: Counting points on K3 surfaces and other arithmetic-geometric objects
Author: Visse, H.D.
Issue Date: 2018-12-18
Keywords: Rational points
K3 surfaces
Serre's problem
Conic bundles
Circle method
Brauer groups
Kummer surfaces
Effective computations
Abstract: This PhD thesis concerns the topic of arithmetic geometry. We address three different questions and each of the questions in some way is about counting how big some set is or can be. We produce heuristics for counting rational points on surfaces given by one diagonal quartic equation. Our results match with experimental data obtained by van Luijk a few years ago. A different result concerns a certain type of conic bundles over low degree hypersurfaces. We count rational points on the base over which the fibre has rational points. We are able to produce asymptotic results where most results in the literature only produce upper bounds. Moreover we investigate the leading constant in this asymptotic formula, matching it up with expected conjectural behaviour that can be found in the literature. Lastly, we study Brauer groups of Kummer surfaces. We give a way to obtain upper bounds for their sizes. Our way is effective (one only needs to use a formula), but the bounds obtained seem not to be sharp. Our method is based on effective versions of Faltings' theorem on finiteness of abelian varieties.
Promotor: Supervisor: Luijk R. van, Stevenhagen P.
Faculty: Faculty of Science
University: Leiden University
Uri: urn:isbn:9789463234122
Handle: http://hdl.handle.net/1887/67532
 

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application/pdf Introduction 295.6Kb View/Open
application/pdf Chapter 1 546.1Kb View/Open
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application/pdf Chapter 4 536.1Kb View/Open Full text at publisher site
application/pdf Bibliography 241.1Kb View/Open
application/pdf Summary in English 224.3Kb View/Open
application/pdf Summary in Dutch 225.0Kb View/Open
application/pdf Acknowledgements_Curriculum Vitae 142.6Kb View/Open
application/pdf Propositions 186.8Kb View/Open

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