K3 ARITHMETIC · Arithmetic of K3 Surfaces
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2007-12-01 → 2008-08-31
- EU contribution
- €125,539
- Participants
- 1
- Scheme
- EIF
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Results in brief
Final Activity Report Summary - K3 ARITHMETIC (Arithmetic of K3 Surfaces)
The project was concerned with the arithmetic of K3 surfaces. K3 surfaces lie at the crossroads of geometry, number theory and physics. The project was also concerned with Diophantine equations which are equations where we seek integral or rational solutions. The study of Diophantine equations is part of the subject arithmetic geometry which brings together number theory and algebraic geometry. Many of the classical Diophantine equations studied over the last 250 years turn out to be K3 surface or related to K3 surfaces. The project made substantial progress on several fronts. One of the most striking results shows (under very mild conditions) that the rational points on diagonal quartic surfaces (whose study goes back to Euler in the 18th Century) are dense; a crude way of saying this is that these classical equations have very many solutions. A second result of the project establishes unexpected and striking links between arithmetic geometry and topology (knot theory in particular). Another result of the project simplifies writing down the equations of genus 2 Jacobians. This is expected to have interesting cryptographic applications.
Data: CORDIS, © European Union
Project objective
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.
Original text from CORDIS.
Participants
- UNIVERSITY OF WARWICK · COVENTRYCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union