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Arithmetic of K3 Surfaces

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.