# 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.