631 research outputs found
On a question of Abraham Robinson's
In this note we give a negative answer to Abraham Robinson's question whether
a finitely generated extension of an undecidable field is always undecidable.
We construct 'natural' undecidable fields of transcendence degree 1 over Q all
of whose proper finite extensions are decidable. We also construct undecidable
algebraic extensions of Q that allow decidable finite extensions
Defining Z in Q
We show that Z is definable in Q by a universal first-order formula in the language of rings. We also present an ∀∃-formula for Z in Q with just one universal quantifier. We exhibit new diophantine subsets of Q like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof for the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for Z in Q, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over Q with many Q-rational points
On the "Section Conjecture" in anabelian geometry
Let X be a smooth projective curve of genus >1 over a field K which is
finitely generated over the rationals. The section conjecture in Grothendieck's
anabelian geometry says that the sections of the canonical projection from the
arithmetic fundamental group of X onto the absolute Galois group of K are (up
to conjugation) in one-to-one correspondence with K-rational points of X. The
birational variant conjectures a similar correspondence where the fundamental
group is replaced by the absolute Galois group of the function field K(X).
The present paper proves the birational section conjecture for all X when K
is replaced e.g. by the field of p-adic numbers. It disproves both conjectures
for the field of real or p-adic algebraic numbers. And it gives a purely group
theoretic characterization of the sections induced by K-rational points of X in
the birational setting over almost arbitrary fields.
As a biproduct we obtain Galois theoretic criteria for radical solvability of
polynomial equations in more than one variable, and for a field to be PAC, to
be large, or to be Hilbertian.Comment: 21 pages, late
An existential 0-definition of F_q[[t]] in F_q((t))
We show that the valuation ring F_q[[t]] in the local field F_q((t)) is
existentially definable in the language of rings with no parameters. The method
is to use the definition of the henselian topology following the work of
Prestel-Ziegler to give an existential-F_q-definable bounded neighbouhood of 0.
Then we `tweak' this set by subtracting, taking roots, and applying Hensel's
Lemma in order to find an existential-F_q-definable subset of F_q[[t]] which
contains tF_q[[t]]. Finally, we use the fact that F_q is defined by the formula
x^q-x=0 to extend the definition to the whole of F_q[[t]] and to rid the
definition of parameters.
Several extensions of the theorem are obtained, notably an existential
0-definition of the valuation ring of a non-trivial valuation with divisible
value group.Comment: 9 page
Definable henselian valuations
In this note we investigate the question whether a henselian valued field
carries a non-trivial 0-definable henselian valuation (in the language of
rings). It follows from the work of Prestel and Ziegler that there are
henselian valued fields which do not admit a 0-definable non-trivial henselian
valuation. We give conditions on the residue field which ensure the existence
of a parameter-free definiton. In particular, we show that a henselian valued
field admits a non-trivial 0-definable valuation when the residue field is
separably closed or sufficiently non-henselian, or when the absolute Galois
group of the (residue) field is non-universal.Comment: 14 pages, revised versio
Free product of absolute Galois groups
The free profinite product of finitely many absolute Galois group is an
absolute Galois group
On the birational section conjecture with local conditions
A birationally liftable Galois section s of a hyperbolic curve X/k over a
number field k yields an adelic point x(s) in the smooth completion of X. We
show that x(s) is X-integral outside a set of places of Dirichlet density 0, or
s is cuspidal. The proof relies on -quotients of for
some open U of X.
If k is totally real or imaginary quadratic, we prove that all birationally
adelic, non-cuspidal Galois sections come from rational points as predicted by
the section conjecture of anabelian geometry. As an aside we also obtain a
strong approximation result for rational points on hyperbolic curves over Q or
imaginary quadratic fields.Comment: Theorem C (and Section 7) of the original version have been deleted
due to a gap in the proof. This is the journal versio
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