130 research outputs found
A topological approach to undefinability in algebraic extensions of
For any subset , consider the set of subfields
which contain a co-infinite subset that is universally definable in such that . Placing a natural topology on the set
of subfields of , we
show that if is not thin in , then is meager in
. Here, thin and meager both mean "small",
in terms of arithmetic geometry and topology, respectively. For example, this
implies that only a meager set of fields have the property that the ring of
algebraic integers is universally definable in . The main
tools are Hilbert's Irreducibility Theorem and a new normal form theorem for
existential definitions. The normal form theorem, which may be of independent
interest, says roughly that every -definable subset of an algebraic
extension of is a finite union of single points and projections of
hypersurfaces defined by absolutely irreducible polynomials.Comment: 24 pages. Introduction has been rewritte
Unlikely intersections in products of families of elliptic curves and the multiplicative group
Let be the Legendre elliptic curve of equation
. We recently proved that, given linearly
independent points on with
coordinates in , there are at most finitely many
complex numbers such that the points satisfy two independent relations on . In
this article we continue our investigations on Unlikely Intersections in
families of abelian varieties and consider the case of a curve in a product of
two non-isogenous families of elliptic curves and in a family of split
semi-abelian varieties.Comment: To appear in The Quarterly Journal of Mathematic
Panorama of p-adic model theory
ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud
Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud
des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud
par une revue de la bibliographie
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