42 research outputs found
Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields
We give a definition, in the ring language, of Z_p inside Q_p and of F_p[[t]]
inside F_p((t)), which works uniformly for all and all finite field
extensions of these fields, and in many other Henselian valued fields as well.
The formula can be taken existential-universal in the ring language, and in
fact existential in a modification of the language of Macintyre. Furthermore,
we show the negative result that in the language of rings there does not exist
a uniform definition by an existential formula and neither by a universal
formula for the valuation rings of all the finite extensions of a given
Henselian valued field. We also show that there is no existential formula of
the ring language defining Z_p inside Q_p uniformly for all p. For any fixed
finite extension of Q_p, we give an existential formula and a universal formula
in the ring language which define the valuation ring
The existential theory of equicharacteristic henselian valued fields
We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of Fq((t))
Definable sets, motives and p-adic integrals
We associate canonical virtual motives to definable sets over a field of
characteristic zero. We use this construction to show that very general p-adic
integrals are canonically interpolated by motivic ones.Comment: 45 page
Definable equivalence relations and zeta functions of groups
We prove that the theory of the -adics admits elimination
of imaginaries provided we add a sort for for each . We also prove that the elimination of
imaginaries is uniform in . Using -adic and motivic integration, we
deduce the uniform rationality of certain formal zeta functions arising from
definable equivalence relations. This also yields analogous results for
definable equivalence relations over local fields of positive characteristic.
The appendix contains an alternative proof, using cell decomposition, of the
rationality (for fixed ) of these formal zeta functions that extends to the
subanalytic context.
As an application, we prove rationality and uniformity results for zeta
functions obtained by counting twist isomorphism classes of irreducible
representations of finitely generated nilpotent groups; these are analogous to
similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald
for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math.
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