Model Theory of the Witt Frobenius, with Emphasis on an Ax-Kochen- Ersov Principle
(Un exposé de Angus MacIntyre au séminaire de
théorie des nombres de Chevaleret le 7 novembre 2005)
Résumé :
We consider the Witt vectors over an algebraically closed field, with
the extra structure of the Witt Frobenius map . Thus, the Witt vectors
are seen as a difference ring, and we study the structure of sets
definable in this difference ring. A similar enterprise has been useful
for diophantine geometry in the case of algebraically closed fields
with automorphism (whose model theory has been used by Hrushovski in
connection with Manin-Mumford). But here the analysis has a different
flavour, because of the lifting of an automorphism of the residue
field. So the work is in fact a very broad generalization of the
Ax-Kochen-Ersov work from the 1960's (since in our case the fixed ring
is the ring of p-adic integers). The 1960's work, and its elaboration
by others, was useful first in connection with a problem of Artin, and
later in connection with p-adic ,and ultimately motivic, integration.
We give analogues of most of the results of Ax-Kochen Ershov in the
Witt Frobenius setting, including decidability results,
quantifier-eliminations in Pas style, passage to characteristic zero
from finite characteristic residue field, and some uniformity in p. I
will give special attention to the analogue of the Ax-Kochen-Ershov
principle that in the limit the p-adics and the power series over the
prime field are the same.
This is joint work with Belair and Scanlon.