# Arithmetic of K3 surfaces (08w5083)

## Organizers

Jean-Louis Colliot-Thélène (Universite Paris-Sud)

Adam Logan (University of Waterloo)

David McKinnon (University of Waterloo)

Alexei Skorobogatov (Imperial College London)

Yuri Tschinkel (Courant Institute NYU and University of Goettingen)

Ronald van Luijk (Universiteit Leiden)

## Description

Arguably the most famous theorem of mathematics is Pythagoras'

theorem. Most people remember it from high school as "a-squared plus

b-squared equals c-squared." Here c is the hypotenuse of a

right-angled triangle, while a and b are the other two sides. Although

this theorem allows the hypotenuse of any right-angled triangle to be

computed, most exercises about this theorem make use of some special

triangles where all sides happen to be whole numbers. The smallest

example is the triangle with sides (3,4,5), but other commonly used

examples are (5,12,13), or (7,24,25), or (8,15,17). These triangles

correspond to the points (3/5,4/5), (5/13,12/13), (7/25,24/25), and

(8/17,15/17) on the unit circle given by x^2 + y^2 = 1. One can prove

that there are in fact infinitely many of these special triangles,

corresponding to infinitely many points with rational coordinates on

the unit circle. Such points are called rational points.

Some mathematicians, often described as number theorists, do not care

much about the triangles, but they are interested in the solutions in

whole numbers of the equation a^2 + b^2 = c^2. These (infinitely many)

solutions have been understood for millennia now. Only a little over

a decade ago, Fermat's Last Theorem was proved, which states that if

we replace the exponent 2 by any other whole number greater than 2,

then all solutions to the equation are trivial in the sense that one

of the variables a, b, and c is equal to zero. Increasing the

exponents in the equation makes the problem significantly harder.

Another way to increase the complexity of the problem is by increasing

the number of variables. Instead of looking for rational points on the

unit circle, we then look for rational points on objects of dimension

2 or greater. These varieties, as they are called, can be classified

and on some of them the rational points are very well understood.

On many varieties, however, they are not. A K3 surface is an example

of such a variety. For example, for some years it was not known

whether the K3 surface given by the equation x^4 + 2y^4 - 4z^4 = 1

has any solutions other than x = 1 or x = -1 and y = z = 0. In fact

there are others, such as x = 1484801/1157520, y = 1203120/1157520,

z = 1169407/1157520, and the solutions that are obtained by taking the

negative of some of the coordinates. It is still not known, however,

whether there are any more solutions. In the last five years the rate

of progress on the understanding of rational points on K3 surfaces

has increased dramatically. This is the first international workshop

to join the forces of all the mathematicians involved in this process.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).