UBC Mathematics Colloquium
Johannes Nicaise
(University of Leuven, Belgium)
Rational points on varieties over a discretely valued field
Fri., Sept. 11, 2009, 3:00pm, MATX 1100
The starting point of this talk is a theorem of Serre's on the
classification of compact p-adic manifolds. Every such manifold is a
disjoint union of n closed unit balls, for a unique value of n in
{0,...,p-1}. Using motivic integration, the ideas of Serre's proof can
be
generalized to algebraic varieties X over a complete discretely valued
field K (for instance, the field of complex Laurent series). In this
way,
one can define the motivic Serre invariant S(X) of X, which is an
element
of a certain ring of virtual varieties over the residue field of K. We
will explain how one can consider S(X) as a measure for the set of
rational points on X, and how this measure admits a cohomological
interpretation by means of a trace formula.
In the first part of the talk, we recall the definitions of p-adic
numbers and p-adic manifolds, and we explain the elementary but elegant
proof of Serre's theorem. In the second part, we develop the basic
notions of motivic integration, and we illustrate the construction of
the
motivic Serre invariant by some examples. In the third part, we explain
the statement of the trace formula, and we give some applications to
complex singularity theory and to arithmetic geometry.