Talk 2:  CM lifting of abelian varieties

Abstract:  If X is an abelian variety over a finite field k then it is known
that for some finite extension k'/k there is a p-adic integer ring R' with residue field k'
and a CM abelian variety A' over Frac(R') with good reduction over k' that is k'-isogenous to X.  It
is natural to ask if the intervention of the extension k'/k and the k'-isogeny are both necessary.
It was shown around 10 years ago by Oort that isogenies are unavoidable,
but there remained open the question of whether one can make a CM-lifting
of some member of the k-isogeny class of X without increasing the residue field beyond k.
Somewhat surprisingly, there turns out to be an arithmetic obstruction (and it
always vanishes for elliptic curves).  We explain this obstruction and illustrate it with some examples.
The main result is that this obstruction to CM lifting without increasing k is the _only_ obstruction.
The proof involves a mixture of class field theory and CM theory.
(This is joint work with Chai and Oort.)