845 research outputs found
An extension of Greenberg's theorem to general valuation rings
We extend Greenberg's strong approximation theorem to schemes of finite
presentation over valuation rings with arbitrary value group, using the
ultraproduct method of Becker, Denef, Lipshitz and van den Dries. As an
application, we prove a closed image theorem (in the strong topology on
rational points) for proper morphisms of varieties over valued fields.Comment: 15 pages, dedicated to Jan Denef; Manuscripta mathematica, 2011. The
final article is avaliable at springerlink.com:
http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s00229-011-0510-
Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
Let be a one-variable function field over a field of constants of
characteristic 0. Let be a holomorphy subring of , not equal to . We
prove the following undecidability results for : If is recursive, then
Hilbert's Tenth Problem is undecidable in . In general, there exist
such that there is no algorithm to tell whether a
polynomial equation with coefficients in \Q(x_1,...,x_n) has solutions in
.Comment: This version contains minor revisions and will appear in Annales de l
Institut Fourie
Sur la définissabilité existentielle de la non-nullité dans les anneaux
16 pages, PDF avec liens hypertexte. Révisé le 18/09/2007. Accepté pour publication dans «Algebra & Number Theory» le 19/10/2007.International audienceOn étudie les anneaux (notamment noethériens) dans lesquels l'ensemble des éléments non nuls est existentiel positif (réunion finie de projections d'ensembles « algébriques »). Dans le cas noethérien intègre, on montre notamment que cette condition est vérifiée pour tout anneau qui n'est pas local hensélien, et qu'elle ne l'est pas pour un anneau local hensélien excellent qui n'est pas un corps. Ces résultats apportent au passage une réponse à une question de Popescu sur l'approximation forte pour les couples henséliens
Elliptic curves and Hilbert's tenth problem for algebraic function fields over real and p-adic fields
Let k be a field of characteristic zero, V a smooth, positive-dimensional,
quasiprojective variety over k, and D a nonempty effective divisor on V. Let K
be the function field of V, and A the semilocal ring of D in K.
In this paper, we prove the Diophantine undecidability of:
(1) A, in all cases;
(2) K, when k is (formally) real and V has a real point;
(3) K, when k is a subfield of a p-adic field, for some odd prime p.
To achieve this, we use Denef's method: from an elliptic curve E over Q,
without complex multiplication, one constructs a quadratic twist E' of E over
Q(t), which has Mordell-Weil rank one. Most of the paper is devoted to proving
(using a theorem of R. Noot) that one can choose f in K, vanishing at D, such
that the group E'(K) deduced from the field extension K/Q(f)=Q(t) is equal to
E'(Q(t)). Then we mimic the arguments of Denef (for the real case) and of Kim
and Roush (for the p-adic case).Comment: 73 pages, LateX2e with hyperref, 1 Postscript figure. v2: Forgotten
assumption added in statement of Theorem 1.1 (the result as stated in the
abstract is correct). v3: accepted for publication in Crelle's Journal; minor
typos corrected, references updated, error corrected in Proposition 2.6.
Actions algébriques de groupes arithmétiques
Nous obtenons un résultat général de finitude pour le H1 de certains schémas en groupes linéaires sur les anneaux de (S-)entiers des corps globaux
- …