This volume contains notes which accompanied the lectures at the tenth annual
Arizona Winter School, held from March 10 to 14, 2007 at the University of Arizona
in Tucson. The Arizona Winter School is an intensive five-day meeting, each year
organized around a different central topic in arithmetic geometry, featuring several
courses by leading and emerging experts (“an annual pilgrimage,” in the words of
one participant). The Winter School is the main activity of the Southwest Center
for Arithmetic Geometry, which was founded in 1997 by a group of seven mathe-
maticians working in the southwest United States, and which has been supported
since that time by the National Science Foundation.
The special character of the Arizona Winter School comes from its format.
Each speaker proposes a project, and a month before the Winter School begins,
the speaker is assigned a group of graduate students who work on the project. The
speakers also provide lecture notes and a bibliography. During the actual school the
speaker and and his or her group of students work every evening on the assigned
project. On the last day of the workshop, the students from each group present
their work to the whole school. The result is a particularly intense and focused five
days of mathematical activity (for the students and speakers alike).
The topic of the Winter School in 2007 was p-adic geometry, and the speakers
were Matthew Baker, Brian Conrad, Kiran Kedlaya, and Jeremy Teitelbaum. Samit
Dasgupta joins Teitelbaum as a co-author. We thank the authors for their hard
work before, during, and after the Winter School. We are also grateful to John Tate
and Vladimir Berkovich, two pioneers of non-archimedean geometry, who agreed
to describe the early history of some of their contributions to the subject. The
anonymous reviewers made numerous valuable comments, and we thank them for
their careful reading of this manuscript. Finally, we are indebted to the other
members (past and present) of the Southwest Center; it is thanks to their efforts
that the Winter School exists in its present form.