# Arithmetic of K3 surfaces (08w5083)

Arriving in Banff, Alberta Sunday, November 30 and departing Friday December 5, 2008

## Organizers

(Universite Paris-Sud)

(University of Waterloo)

(Imperial College London)

(Courant Institute NYU and University of Goettingen)

(Universiteit Leiden)

## Objectives

Understanding Diophantine equations is one of the fundamental goals of
mathematics. Algebraic geometry has proved to be indispensable in the
study of Diophantine problems. It is therefore no wonder that
throughout history the geometric complexity of the Diophantine
problems in focus has been increasing steadily. While the arithmetic
of curves has been studied for a long time now, only fairly recently
has there been substantial progress on that of higher-dimensional
varieties. Naturally, this started with the easier varieties, such as
rational and abelian varieties. K3 surfaces, where many basic
problems are still wide open, are the next step in complexity.

In the last five years the rate of progress on the arithmetic of K3
surfaces has increased dramatically. However, not a single
international meeting has been held to join the forces of the people
involved. It is of the utmost importance to hold a workshop to
combine the many lines of research in this new area. The big open
problems can only be tackled by combining different strengths, both
computational and theoretical. Everyone mentioned in the overview of
the subject and many others have expressed very strong interest. Among
these people are

David McKinnon (University of Waterloo),
Nils Bruin (Simon Fraser University),
Noriko Yui (Queen's University),
Hershy Kisilevsky (Concordia, Montr'eal),
Ronald van~Luijk (PIMS/UBC/SFU),
Yuri Tschinkel (University of G"ottingen),
Sir Peter Swinnerton-Dyer (Cambridge),
Alexei Skorobogatov (Imperial College),
Brendan Hassett (Rice University),
Jean-Louis Colliot-Th'elene (Paris-Sud Orsay),
Bjorn Poonen (UC Berkeley),
Helena Verrill (Louisiana State University),
Lucia Caporaso (University of Rome, 3),
Arthur Baragar (University of Nevada, Las Vegas),
Fedor Bogomolov (New York University),
Michael Stoll (International University Bremen),
Andrew Kresch (University of Z"urich),
Jordan Ellenberg (University of Wisconsin, Madison),
Jasper Scholten (University of Leuven),
Matthias Sch"utt (University of Hannover),
Patrick Corn (University of Georgia, Athens),
David Harari (Paris-Sud Orsay),
Olivier Wittenberg (Rice University),
Damiano Testa (University of Rome),
and Martin Bright (Heilbronn Institute).

With many young people (Paola Argentin, Damiano Testa, Abhinav Kumar,
Martin Bright, Olivier Wittenberg, Adam Logan, Ronald van~Luijk)
working on K3 surfaces nowadays, it is time for the very first
workshop dedicated to the arithmetic of K3 surfaces. Such a workshop
will undoubtedly have an immense impact on the field. The fields of
specialization of the participants we intend to invite include the
following.

* Modularity of K3 surfaces

* Potential density of rational points

* Brauer-Manin obstructions

* Weak approximation

* Growth of the number of rational points of bounded height

* Computability of the Picard group

* Applications to curves

* Universal torsors

* K3 surfaces in positive characteristic

* Enriques surfaces

Besides the people mentioned above we have a list of quite a few more
people that we believe will be interested but whom we have not yet
contacted. We also intend to have several more young participants,
both new postdocs and graduate students. For this reason, and because
the participants all come from different backgrounds, we will start
with several survey lectures on the topics mentioned above. The
participants will then be able to form small groups to focus on more
specialized issues.

Banff would be a perfect place for several reasons. Given the number
of highly active people in the field that are in their postdoc stage
with little funding, it would be extremely useful to have their
expenses paid. The size of the accommodation at Banff is also
perfect. In light of the unfortunate fact that no dates will suit
everybody, a workshop for 40 participants will allow all experts in
the field to attend, as well as postdocs and eager graduate students.

The name `K3 surfaces'' refers to the three algebraic geometers
Kummer, K"ahler and Kodaira, but also alludes to the mountain peak
K2, which had recently been climbed for the first time when the name
was given during the 1950s. We expect that this conference in Banff
will have a tremendous impact on the arithmetic of K3 surfaces. In
this way K3 surfaces will once again be linked to the mountains.