Arithmetic of K3 Surfaces
Start date: Dec 1, 2007,
End date: Aug 31, 2008
Luijk's research is on explicit methods in higher-dimensional arithmetic algebraic geometry, motivated by Diophantine problems. In particular, he works on K3 surfaces, which are of interest to algebraic geometers, number theorists and theoretical physicist s. In his thesis, Luijk solved two explicit open Diophantine problems. He also developed a method to bound the rank of the Neron-Severi group of a K3 surfaces. He has constructed K3 surfaces over the rationales with infinitely many rational points and Neron-Severi rank 1. This answers a question by Swinnerton-Dyer and disposes of an old challenge attributed to Mumford. Recently he has found the first known examples of K3 surfaces with trivial automorphism group.For the project it intended to tackle four problems:Problem 1: Give an algorithm to compute the Neron-Severi group of K3 surfaces.Problem 2: Formulate a suitable Manin-type conjecture for K3 surfaces, linking the distribution of rational points to the Neron-Severi group. Gather theoretical and experimental evidence for this.Problem 3: Give necessary and sufficient criteria for the failure of the Hasse principle to be accounted for by the Brauer-Manin obstruction in terms of the Neron-Severi group.Problem 4: The homogeneous spaces for 2-descent on genus 2 curves have K3 surfaces as quotients.Investigate the implications of our work for the arithmetic of genus 2 curves and their Shafarevich-Tate groups. At Warwick Luijk will be able to draw on the expertise of Reid in algebraic geometry, Kresch in abstract arithmetic geometry and Siksek in explicit arithmetic geometry and Diophantine equations.
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