The Néron component series of a degenerating family of abelian
varieties
Abstract: Let K be a complete discretely valued field and
let
A be an abelian K-variety. In this talk I will discuss the
Néron component series of A. This is a formal power series in
Z[[T]] which measures how the number of connected components
of the special fiber of the Néron model of A varies under
tame extensions of A.
In case A is wildly ramified, it is particularly challenging
to describe the properties of this series. I will present some
results for Jacobians and abelian varieties with potential
multiplicative reduction, and discuss a few open problems in this setting.
This is joint work with Johannes Nicaise.