Division-ample sets and the Diophantine problem for rings of integers
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 727-735.

We prove that Hilbert’s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.

Nous demontrons que le dixième problème de Hilbert pour un anneau d’entiers dans un corps de nombres K admet une réponse négative si K satisfait à deux conditions arithmétiques (existence d’un ensemble dit division-ample et d’une courbe elliptique de rang un sur K). Nous lions les ensembles division-ample à l’arithmétique des variétés abéliennes.

DOI: 10.5802/jtnb.516
Cornelissen, Gunther 1; Pheidas, Thanases 2; Zahidi, Karim 3

1 Mathematisch Instituut Universiteit Utrecht Postbus 80010 3508 TA Utrecht, Nederland
2 Department of Mathematics University of Crete P.O. Box 1470 Herakleio, Crete, Greece
3 Equipe de Logique Mathématique U.F.R. de Mathématiques (case 7012) Université Denis-Diderot Paris 7 2 place Jussieu 75251 Paris Cedex 05, France Adresse actuelle: Departement Wiskunde, Statistiek & Actuariaat Universiteit Amtwerpen Prinsstraat 13 2000 Antwerpen, België
@article{JTNB_2005__17_3_727_0,
     author = {Cornelissen, Gunther and Pheidas, Thanases and Zahidi, Karim},
     title = {Division-ample sets and the {Diophantine}  problem for rings of integers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {727--735},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {3},
     year = {2005},
     doi = {10.5802/jtnb.516},
     mrnumber = {2212121},
     zbl = {05016583},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.516/}
}
TY  - JOUR
AU  - Cornelissen, Gunther
AU  - Pheidas, Thanases
AU  - Zahidi, Karim
TI  - Division-ample sets and the Diophantine  problem for rings of integers
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
DA  - 2005///
SP  - 727
EP  - 735
VL  - 17
IS  - 3
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.516/
UR  - https://www.ams.org/mathscinet-getitem?mr=2212121
UR  - https://zbmath.org/?q=an%3A05016583
UR  - https://doi.org/10.5802/jtnb.516
DO  - 10.5802/jtnb.516
LA  - en
ID  - JTNB_2005__17_3_727_0
ER  - 
%0 Journal Article
%A Cornelissen, Gunther
%A Pheidas, Thanases
%A Zahidi, Karim
%T Division-ample sets and the Diophantine  problem for rings of integers
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 727-735
%V 17
%N 3
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.516
%R 10.5802/jtnb.516
%G en
%F JTNB_2005__17_3_727_0
Cornelissen, Gunther; Pheidas, Thanases; Zahidi, Karim. Division-ample sets and the Diophantine  problem for rings of integers. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 727-735. doi : 10.5802/jtnb.516. http://www.numdam.org/articles/10.5802/jtnb.516/

[1] J. Cheon, S. Hahn, The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve. Acta Arith. 88 (1999), no. 3, 219–222. | MR | Zbl

[2] G. Cornelissen, Rational diophatine models of integer divisibility, unpublished manuscript (May, 2000).

[3] G. Cornelissen, K. Zahidi, Topology of Diophantine sets: remarks on Mazur’s conjectures. Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), 253–260, Contemp. Math. 270, Amer. Math. Soc., Providence, RI, 2000. | MR | Zbl

[4] J. Cremona, mwrank, www.maths.nott.ac.uk/personal/jec/, 1995-2001. | MR

[5] M. Davis, Y. Matijasevič, J. Robinson, Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution. Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), pp. 323–378. | MR | Zbl

[6] J. Denef, Hilbert’s tenth problem for quadratic rings. Proc. Amer. Math. Soc. 48 (1975), 214–220. | MR | Zbl

[7] J. Denef, Diophantine sets of algebraic integers, II. Trans. Amer. Math. Soc. 257 (1980), no. 1, 227–236. | MR | Zbl

[8] J. Denef, L. Lipshitz, Diophantine sets over some rings of algebraic integers. J. London Math. Soc. (2) 18 (1978), no. 3, 385–391. | MR | Zbl

[9] T. Pheidas, Hilbert’s tenth problem for a class of rings of algebraic integers. Proc. Amer. Math. Soc. 104 (1988), no. 2, 611–620. | MR | Zbl

[10] T. Pheidas, K. Zahidi, Undecidability of existential theories of rings and fields: a survey, in: “Hilbert’s tenth problem: relations with arithmetic and algebraic geometry” (Ghent, 1999). Contemp. Math. 270, Amer. Math. Soc. (2000), 49–105. | MR | Zbl

[11] B. Poonen, Using elliptic curves of rank one towards the undecidability of Hilbert’s tenth problem over rings of algebraic integers. Algorithmic Number Theory (eds. C. Fieker, D. Kohel), 5th International Symp. ANTS-V, Sydney, Australia, July 2002, Proceedings, Lecture Notes in Computer Science 2369, Springer-Verlag, Berlin, 2002, pp. 33-42. | MR | Zbl

[12] H. Shapiro, A. Shlapentokh, Diophantine relations between algebraic number fields. Comm. Pure Appl. Math. XLII (1989), 1113-1122. | MR | Zbl

[13] A. Shlapentokh, Hilbert’s tenth problem over number fields, a survey, in: “Hilbert’s tenth problem: relations with arithmetic and algebraic geometry” (Ghent, 1999). Contemp. Math. 270, Amer. Math. Soc. (2000), 107–137. | MR | Zbl

[14] A. Shlapentokh, Extensions of Hilbert’s tenth problem to some algebraic number fields. Comm. Pure Appl. Math. XLII (1989), 939–962. | MR | Zbl

[15] J.H. Silverman, The arithmetic of elliptic curves. Graduate Texts in Math. 106, Springer-Verlag, New York, 1986. | MR | Zbl

[16] D.  Simon, Computing the rank of elliptic curves over number fields. LMS J. Comput. Math. 5 (2002), 7–17. | MR | Zbl

[17] M. Stoll, Hyperelliptic curves MAGMA-package, www.math.iu-bremen.de/stoll/magma/.

Cited by Sources: