We prove that Hilbert’s Tenth Problem for a ring of integers in a number field has a negative answer if satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over ). 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 admet une réponse négative si satisfait à deux conditions arithmétiques (existence d’un ensemble dit division-ample et d’une courbe elliptique de rang un sur ). Nous lions les ensembles division-ample à l’arithmétique des variétés abéliennes.
DOI: 10.5802/jtnb.516
Author's affiliations:
@article{JTNB_2005__17_3_727_0,
author = {Gunther Cornelissen and Thanases Pheidas and Karim Zahidi},
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},
zbl = {05016583},
mrnumber = {2212121},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.516/}
}
TY - JOUR 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 - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.516/ UR - https://zbmath.org/?q=an%3A05016583 UR - https://www.ams.org/mathscinet-getitem?mr=2212121 UR - https://doi.org/10.5802/jtnb.516 DO - 10.5802/jtnb.516 LA - en ID - JTNB_2005__17_3_727_0 ER -
Gunther Cornelissen; Thanases Pheidas; Karim Zahidi. 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. https://jtnb.centre-mersenne.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: 1683630 | Zbl: 0933.11029
[2] G. Cornelissen, Rational diophatine models of integer divisibility, unpublished manuscript (May, 2000).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: 1802017 | Zbl: 0982.14014
[4] J. Cremona, mwrank, www.maths.nott.ac.uk/personal/jec/, 1995-2001. | MR: 1835853
[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: 432534 | Zbl: 0346.02026
[6] J. Denef, Hilbert’s tenth problem for quadratic rings. Proc. Amer. Math. Soc. 48 (1975), 214–220. | MR: 360513 | Zbl: 0324.02032
[7] J. Denef, Diophantine sets of algebraic integers, II. Trans. Amer. Math. Soc. 257 (1980), no. 1, 227–236. | MR: 549163 | Zbl: 0426.12009
[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: 518221 | Zbl: 0399.10049
[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: 962837 | Zbl: 0697.12020
[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: 1802009 | Zbl: 0981.03013
[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: 2041072 | Zbl: 1057.11068
[12] H. Shapiro, A. Shlapentokh, Diophantine relations between algebraic number fields. Comm. Pure Appl. Math. XLII (1989), 1113-1122. | MR: 1029120 | Zbl: 0698.12022
[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: 1802010 | Zbl: 0994.03001
[14] A. Shlapentokh, Extensions of Hilbert’s tenth problem to some algebraic number fields. Comm. Pure Appl. Math. XLII (1989), 939–962. | MR: 1008797 | Zbl: 0695.12020
[15] J.H. Silverman, The arithmetic of elliptic curves. Graduate Texts in Math. 106, Springer-Verlag, New York, 1986. | MR: 817210 | Zbl: 0585.14026
[16] D. Simon, Computing the rank of elliptic curves over number fields. LMS J. Comput. Math. 5 (2002), 7–17. | MR: 1916919 | Zbl: 1067.11015
[17] M. Stoll, Hyperelliptic curves MAGMA-package, www.math.iu-bremen.de/stoll/magma/.
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